Questions & Answers

With the recent publication of PHYSICS IS... there are now three Ask the Physicist books! Click on the book images below for information on the content of the books and for information on ordering.

QUESTION:
When I carry my cell phone (iPhone 7) in my left pocket and it vibrates, I feel the vibe on my right hand, while my left hand feels nothing. It happens quite often and I even check to see inside my right pocket (or in my tote bag on my right shoulder) I fear of having misplaced my cell there. What does the trick?

ANSWER:
This is not really so surprising. The brain is divided into two halves and there is some crossing of nerve impulses sometimes. I used to hear and feel a thumping in my right ear when I tapped my left big toe. It seems to not do that anymore.

QUESTION:
If you take out the spaces between all the atoms of the nuclei in your body, would you end up half the size of a flea but still weigh the same?

ANSWER:
Technically, there are no spaces between atoms in your body. I think what you are asking is that if we compressed all the matter in your body (as happens in a neutron star) to nuclear density, how big would it be? Suppose your body has a mass of 100 kg. The density of nuclear matter is about 2x10¹⁷ kg/m³, so your volume would be about 10²/2x10¹⁷=5x10^-16 m³. So, if you are a cube, your dimensions would be ³√5x10^-16=8x10^-6 m≈10 microns. A flea has a size of about 1 mm=1000 microns, one hundred times bigger than the compressed you. And, yes, you would still have a mass of 100 kg.

QUESTION:
If we are looking back in time through telescopes now at stars and galaxies, would it be possible to calculate the distance required away from earth in order to look back and see a moment in history here on earth, theoretically ?

ANSWER:
Of course, but you could never get there in time to see it. An alien who was 5000 light years from earth would see, if he had a good enough telescope, the earth as it was 5000 years before he was looking.

QUESTION:
Since even light cannot escape the gravitational pull of a black hole, does this mean that the theoretical particle, known as the graviton, have the potential to move faster than the photon?

ANSWER:
Sorry this took so long, but I had to consult with an astrophysicist. The best answer he could come up with is at this link.

QUESTION:
Attached is a picture culled from the internet on an object rolling down a plane. I understand everything except how to explain and draw the forces to my AP physics students. Should this look like a box sliding down the plane? with mgsintheta down the plane and mgcostheta as the normal force and mg down and friction up? Or are some of these properly represented as torques? Could you please draw how this should be properly represented?

ANSWER:
I have not included your picture because there is too much stuff on it (the velocity, the R, the Rω, etc. all distract from the crux of the problem, in my opinion). I have redrawn the problem showing some axially symmetric object of radius R, mass m rolling down an incline of angle θ without slipping. There are only three forces acting on the object, the frictional force f and the normal force N, both of which act at the point or line of contact, and the weight mg which acts at the center of mass of the object. There are three unknowns, N, f, and a, the acceleration of the center of mass. To solve for these unknowns, we must apply Newton's laws, both translational and rotational (ΣF=ma and Στ=Iα); here I is the moment of inertial about the axis about which torque is calculated and α is the angular acceleration about that axis. You are a teacher, so I will skip a lot of the details which you can fill in easily. From ΣF=ma I find N=mgcosθ and ma=mgsinθ -f. The first equation nails N, one of our unknowns, the second is one equation for the two remaining unknowns, so we are not done. So the only place to get a second equation is the rotational form of Newton's second law; I think this is the trickiest part of this problem. I find many students have an inclination to choose the symmetry axis to sum torques; they have been told, when they learned statics of non point objects, that you can choose any axis you like, but this is not a statics problem and there is a very important constraint—Newton's laws are not valid in an accelerating system, and the center of the rolling object is accelerating. (See correction below!) So Στ=Iα yields Iα=Ia/R=mgRsinθ, or a=(mR²/I)gsinθ; knowing a, then f=m(gsinθ-a) But, what is I? It isn't what you look up in a book because usually what is listed is the moment of inertia about an axis passing through the center of mass (I_cm). Here the axis is a distance R from the center of mass, so using the parallel axis theorem, I=I_cm+mR².

I will do one example, a solid sphere where I_cm=2mR²/5, so I=7mR²/5, so a=5gsinθ/7 and f=2mgsinθ/7. If you sum torques about the center of mass, you get the wrong answer.

CORRECTION:
OK, I slipped up here, albeit in a fairly minor way. While it is true that Newton's laws are not correct in an accelerating frame of reference, there is one exception: the sum of the torques about an axis passing through the center of mass is indeed I_cmα. A little detail I had forgotten about classical mechanics. Let me work that out for my example: I_cmα=2maR/5=fR, so, using the result above, f=2ma/5=m(gsinθ-a) or a=5gsinθ/7. Then it easily follows that f=2mgsinθ/7. But you should try to sum torques about an axis halfway between the point of contact and the center of mass—you will not get the right answer.

Nevertheless, it is important that your students be careful in choosing the axis. I have edited the original answer somewhat.

QUESTION:
How massive is the largest black hole ever discovered?

ANSWER:
See the Wikepedia article List of Most Massive Black Holes. This is the most up-to-date information I could find.

QUESTION:
Does the core of the earth spin? If is does how does it spin faster than the mantel with the viscosity of magma and the 4+ billions years old the earth is? I just don't understand what I am being told.

ANSWER:
The core of the earth is composed mainly of iron and is comprised of an inner core which is solid and an outer core which liquid. Apparently, from what I understand from an article I found, recent research indicates that the outer core rotates with the same speed as the mantle (once a day) but the inner core's rotational speed is slightly faster by about 0.3-0.5 degrees per year.

QUESTION:
If I dropped a circular, triangular, and square parachute from the same height which will land first assuming the parachuted object's weight is the same and the surface area is the same for all three parachutes?

ANSWER:
I am assuming that your parachutes are flat, horizontal plates and that drag forces due to the load and lines attaching the load to the parachute are negligible or at least the same. The drag coefficients C_D for the three shapes, determined experimentally, are shown in the figure. The drag force is given by F_D=½ρv²C_DA where ρ is the density of the air, v is the speed of descent in still air, and A is the area. Therefore the drag force is least on the circular parachute so it reaches the ground first. (A proviso is that the values of the drag coefficient depend on a quantity called the Reynolds number which depends on the velocity. However, the values given are for low Reynolds numbers which are suitable for parachutes.)

QUESTION:
How did Einstein get the idea that the speed of light must be constant for all observer?

ANSWER:
Einstein's original paper on special relativity, On the Electrodynamics of Moving Bodies, states two main problems in electromagnetic theory at the end of the 19th century:

The first paragraph states, essentially, that Maxwell's equations will not have the same form in a frame of reference moving with constant velocity relative to a frame where they are the laws governing electromagnetism if you transform them using Galilean transformations.*
The beginning of the second paragraph acknowledges "…the unsuccessful attempts to discover any motion of the earth relatively to the 'light medium' [luminiferous æther, e.g. the Michaelson-Morley experiment]…"

Einstein had a strongly-held philosophical belief that, for something to be a true law of physics, it must be the same in all reference frames—regardless of where you are or how you are moving, the laws of physics for you are the same as for anybody else in the universe; this is the principle of relativity. Since he believed Maxwell's equations to be laws of physics, he concluded that Galilean relativity was not correct and this led to his postulate that the speed of light was independent of the motion of the source or the observer. To my own mind, I believe that the constancy of the speed of light is not a necessary postulate, rather a consequence of the principle of relativity applied to Maxwell's equations.

*A Galilean transformation is, for example, observing a car moving 60 mph east relative to the ground to be moving 100 mph east if you are in a car moving 40 mph west relative to the ground.

QUESTION:
What type of impact will cause the metal in the car to "splash" (eliminating all hope of a simple conservation of momentum calculation) should the New Horizons interplanetary space probe hit your car?

ANSWER:
The only thing which prevents linear momentum from being conserved is an external force. If there are no forces other than those exerted by the car and the probe (or pieces thereof), linear momentum is conserved. For example, if the collision happens in empty space with the probe of mass M moving with velocity V in a frame where the car is at rest, the total momentum of all the pieces will be MV after the collision or any time during the collision. What will not be conserved is the kinetic energy ½MV².

QUESTION:
In a recent answer, you mention the Cosmic Microwave Background and say that it "provides a frame of reference for the whole universe". It was my understanding that one of the underpinnings of the basic theory of relativity is that there CANNOT be such a universal frame of reference (UFOR). I assume you're not claiming Relativity has been disproven (well, shown to be a special case of something else), so can you explain the difference between the frame of reference meant for relativity and the CMB as you described it?

ANSWER:
Good question! Special relativity rests on the assumption that the laws of physics must be the same in all inertial frames (the principle of relativity); no frames are ruled out as long as they are inertial. In other words, no one frame is better than all the rest. But suppose that we find a frame of reference which contains a "gas of photons" which are all moving in random directions (average velocity of all these photons is zero) and this "gas" pervades the whole universe. Then, if our frame happens to be moving with some velocity v through this gas, we will see that gas having an average velocity -v which would be determined by looking at the doppler shift of the photons; this is analogous to feeling a wind on your face when you run in still air. Having found such a frame allows us to specify an "absolute velocity" which, before now was not possible. What makes this frame special is the fact that, because it is a remnant of the big bang, it does pervade the whole universe.

QUESTION:
I was curiouse at which height would a 6kg turkey have to fall for it to be perfectly cooked when it hits the ground?

ANSWER:
It is very difficult to do a quantitative calculation because the air drag (which would cause heating) changes with altitude as the air gets less and less dense. I doubt that there is any height where this could work because, giving it a high enough speed to compensate for the short time (keep in mind that it takes several hours to roast a turkey at 350⁰F) of fall would result in the outside being burnt and the inside being raw. It might be fun to do a couple of very rough calculations just to get an order of magnitude feeling for what you are interested in.

I will approximate the turkey to have the properties of water, in particular that its specific heat is C=4.186 J/gram⋅⁰C; so the energy which will raise the temperature of a M=6000 gm turkey is E=MCΔT=6000x4.186x56=1.41x10⁶ J. (70⁰F→170⁰F is 21⁰C→77⁰C, hence ΔT=56⁰C.) The change in potential energy in falling from a height h is V=Mgh=6x9.8h=59h, so if the turkey falls with constant velocity the potential energy lost goes into heat. If the heat is 1.41x10⁶ J, the height is h=1.41x10⁶/59=24,000 m. The atmospheric pressure at this altitude is only about 3% what it is at sea level; so it is clear that, if your turkey is given any velocity at this altitude, it will not continue moving with constant velocity. (Whereas, if the pressure were atmospheric and you gave the turkey a speed equal to the terminal velocity it would continue moving with that speed as it fell.) If you dropped it from very far away (essentially infinitely far), it would arrive at the upper atmosphere going nearly the escape velocity, 11,200 m/s. At this speed, which is faster than the reentry speed of the shuttle which needed special ceramic tiles to shield against burning up, the turkey would be burned to a crisp, probably would not reach ground before totally burnt. Although there would be some speed of entry to the atmosphere where the turkey would absorb 1.41x10⁶ J, it would almost certainly not be "perfectly cooked".

QUESTION:
In a nuclear fusion reaction hydrogen atoms combine to create a helium atom. But in a hydrogen atom there are no neutrons, then where does the neutrons in helium nucleus come from ?

ANSWER:
There is more than one answer to this question. We can look first about how hydrogen (with no neutrons, just one proton) becomes helium (two protons, two neutrons) in a star like our sun. As shown by the figure, it begins with pairs of protons which interact and fuse in a reaction which results in one deuteron (heavy hydrogen, one proton and one neutron), one positron (a positively charged electron), and one neutrino. The deuterons thus created now fuse with another proton to create a light isotope of helium, ³He (one neutron, two protons), and also a gamma ray. Two of these ³He now fuse, throwing off two protons in the process and one alpha particle (⁴He, two protons and two neutrons). The two thrown-off protons are now free to continue the process, so this chain, called the proton-proton cycle, is self-sustaining until the star runs out of hydrogen which ends the life of the star.

For more earth-bound hydrogen fusion, like fusion reactors or hydrogen bombs, single-step reactions are sought. The most frequently used is the deuterium-tritium reaction, involving the fusion of one deuteron and one triton (³H, one proton and two neutrons) shown in the other figure.

QUESTION:
If time ticks at 1 second per second in a 1g gravity at the earths surface. Then my question is would a clock tick twice as fast 2 second per second if the earth surface was 0.5g.

ANSWER:
Most certainly not! The expression for the time t elapsed in the gravitational field of a nonrotating spherically symmetric mass M is t=t₀/√(1-δ) where δ=2GM/(Rc²)=2aR/c²; here, R is the distance from the center, t₀ is the time elapsed when R=∞, G is the universal gravitational constant, M is the mass of the sphere, and c is the speed of light, and a is the local acceleration due to gravity. If a=g and R=6.4x10⁶ m (earth radius), it is easy to calculate that δ=1.39x10^-9. Now, t_g/t_½g=[√(1-½δ)]/[√(1-δ)]; using the binomial expansion (1+δ)^x≈1+xδ+… if δ<<1, we finally get t_½g≈t_g(1+¼δ)=t_g(1+3.5x10^-10), quite a bit different from t_½g=2t_g!

QUESTION:
Why does steam rise from the cooking pot when I turn off the heat after the water starts boiling?

ANSWER:
Water evaporates from the surface and the higher the temperature, the faster it evaporates. The higher the rate of evaporation, the more likely it is that individual evaporated molecules will coalesce with others to form droplets of water large enough to be seen. That is what steam is and what clouds are. But steam can be observed even at low temperatures under the right conditions, for example the morning mist on the surface of a lake or fog. The water in the cooking pot, if you observe it carefully, will begin to steam before it starts boiling. So it should be no surprise to find that it still steams after you turn off the heat.

QUESTION:
what makes my radiometer occasionally accelerate wildly when bumped? I can't duplicate the action but have seen it a few times. It has happened around a desk lamp with an LED bulb. Could it be part of an electrical emission around the radiometer? I was allays taught that the photons are what push the vanes

ANSWER:
Well, as it turns out, photon pressure is not what makes the radiometer turn. You can learn about radiation pressure by reading an old answer and the links it contains. I do not know what "accelerate wildly" means in this context but, since you mentioned "bumped", mechanical bumping probably caused unexpected behavior of the radiometer.

QUESTION:
I wanted to ask a question regarding airfoils. When looking at videos online about them, the instructors say that airfoils work because the top of the wing turns air down due to the coanda effect and the downturned air also turns the air under the wing down to create lift. I think I understand the law applied here to be with every action there is an equal and opposite reaction however, with the explanation they gave, it seemed like they were saying that the air on the top that was turned down was the only thing that is turning the air under the wing down. I that true? I would assume that, even though that does happen, the lift generated from the bottom of the actual wing deflecting the air down was more and more important than the coanda effect on the top. The video also said that disrupting the flow on top of the wing through too high of angle of attack would cause it to stall, but wouldn't the actual reason being that the bottom of the wing converts too much lift into drag by changing the opposing force that is lifting it from mostly up to mostly back? There was another source that said that there was a vortex at the back of the wing that pushed the air at the bottom of the wing so that the net force would cancel out a bit, slowing the air down and making a pressure difference to lift the wing. What is that about?

ANSWER:
This question violates the site groundrule for "single, concise, well-focused questions". I can tell you that what is learned in elementary physics courses that airplanes get lift from the Bernouli effect of lower pressure due to higher velocity of air on the top of a wing is a relatively small part of how an airplane is able to fly. As you seem to refer to in your question, the main thing is that air leaves the wing with a downward component of its velocity; this means that the wing exerted a force down on that air which means, via Newton's third law, that the air exerted a force up on the airplane. To quote Wolfgang Langewiesche, author of Stick and Rudder, the private pilot's bible on the art of flying, "The main fact of all heavier-than-air flight is this: the wing keeps the airplane up by pushing the air down."

QUESTION:
How many light-years or parsecs away from 2018 is 1961? The solar system, the earth and the galaxy would have been in a different part of the universe. Do we know the direction of where our galaxy, solar system used to be before we arrived here in the present moment? How many miles has the Earth traveled in space in fives minutes? Including the movement of the solar system and galaxy moving through the vacuum? Millions of parsecs? Billions?

ANSWER:
It never makes sense to talk about velocity of something without specifying the frame of reference relative to which the velocity is measured. And when you are talking about the earth, the motion is very complicated because it rotates about its center (about 0.5 km/s), revolves around the sun (about 30 km/s), and the solar system revolves around the center of the Milky Way galaxy (about 220 km/s). What about the motion of the center of the galaxy? Until recently, you could only measure that speed relative to another object outside our galaxy, e.g., another galaxy or the center of our local cluster of galaxies. However, the discovery of the cosmic microwave background (CMB) has now given us a reference frame which seems to define the reference frame of the whole universe. Very careful measurements and data analyses from the satellite COBE have shown that our galaxy moves with a speed of about 580 km/s relative to the CMB.

So, what can we do to answer your question which is, essentially, how far does the earth move in 57 yr=1.8x10⁹ s? (Your 5 minute question is essentially impossible to answer since it depends on the time of year. Over years the orbital motion averages out to zero.) Relative to the center of the galaxy, the distance is D=220x1.8x10⁹ km= 4x10¹¹ km=0.013 pc. The center of the galaxy in that time will move 580x1.8x10⁹ km=10¹² km=0.032 pc. But I do not know the relative directions of these two velocities, the sum of which would be the average velocity of the solar system over these 57 years; this speed would be anything between 360 and 800 km/s. However, in my researching this question I found that the speed of sun relative to the CMB is about 370 km/s, so the final answer is that the earth moved about 370x1.8x10¹¹=6.7x10¹¹ km=0.022 pc. It is interesting to me that, at this time, the sun and the center of the galaxy are moving in nearly opposite directions (370 km/s compared to 360 km/s); since the time for the sun to go all the way around the galaxy is about 225 million years, in about 110 million years our speed relative to the CMB will be about 800 km/s.

Keep in mind that I am not an astronomer/astrophysicist/cosmologist! I have gathered these data as best I can.

QUESTION:
I have a potentially stupid question about pressure in a vacuum. So I just found out that in a vacuum chamber, microns can be used for measuring vacuum pressure because it's representative of a DISTANCE- the distance being the displacement of columns of Mercury. In understanding Leak Up Rate tests in a vacuum furnace, the Leak Up Rate is a function of Difference of Vacuum Levels/Elapsed time. My question is, does the displacement of Mercury happen faster with more time? IE, does the displacement of Mercury gain momentum with extended periods of time under vacuum? Or, IE, is it exponential instead of linear?

ANSWER:
Very likely there is not a column of mercury being used to measure pressure in your chamber. There is probably some mechanical device using strain gauges which is calibrated in mm or μm of Hg. So, you do not have to worry about what a column of mercury is going to do as the pressure changes. 1 μm (micron of Hg)=0.133322 Pa (N/m²)=1.93368x10^-5 PSI=1.31579x10^-6 atm.

QUESTION:
After a math class yesterday (homeschool) , my son and I were discussing an article I read a few days earlier regarding a exploded star that went super nova about 150 million years ago. The event bathed the earth in a stream of high energy cosmic rays for a couple of months and possibly caused an extinction level event that effected ocean life of the day.

As the conversation progressed, we considered the implications and began to wonder if it was possible to shield against cosmic radiation? To the best of my knowledge, I explained that no substance I know of can block cosmic radiation and that the Earth's magnetic field would have little effect either. The only effects I know of are when they impact atoms in the atmosphere or the earth. If I understand this, if it doesn't hit something fairly dense, it just keeps going, understandable given the space between atoms.

I have read about the gravitational lensing effects on light by massive objects like neutron stars and galaxies that bend light around them. I have also read about the possibility of warping space as a method space travel by changing the shape of space in front of and behind a craft so that it "falls" in a particular direction.

While we were discussing these effects, he looked at me and said something to the effect, "Maybe we could bend space and make them go around?". To me, this was a pretty profound leap in my book. So I decided to ask physicist if he could be right.

Please forgive the length of this post before asking our question. I wanted to provide sufficient context before asking. So, here it is. Is it possible to make cosmic rays go around something, either by gravitational or microwave warping of space?

B.T.W. I really really enjoy your web site. It is awesome!

ANSWER:
I am not a paleontologist, so I had to do a little research on mass extinctions. I discovered that there is no known mass extinction 150 million years ago. In addition, no known extinction is associated with a nearby supernova. (The timeline here is clickable for an enlarged picture.) I would conclude that the article which you read is a "fringe theory" of mass extinctions, not a mainstream fact.

But, I can speak to your main question and comment on some of your statements. Cosmic radiation refers to a broad range of radiations but mostly protons and light nuclei; but also it could refer to neutrinos, electrons, positrons, antiprotons, gamma rays, x-rays, and others. What is not the case, though, is that "no substance…block cosmic radiation"; just about any cosmic radiation can be blocked by a dense solid or liquid. Neutrinos are the exception and most pass all the way through the earth without any interactions. You may know many neutrino experiments are performed in deep abandoned mines to block out all other radiation. The atmosphere also is very effective since a single energetic proton will strike an atom and many other particles, all with less energy than the original proton, will form a "shower" of particles. And magnetic fields are not very effective deflecting the very highest energy charged particles but are known to be an effective shield for lower energy particles.

Finally, let's address the question of warping of space as a way to deflect the rays. The only way to warp space that I know of is to have an enormous mass nearby. The earth already warps the space around it which would cause a particle passing close by, but not on a collision course with earth, to be deflected a tiny bit (viz. gravity). But putting a star sized object near enough the earth to "shield" us would obviously be a catastrophic thing to do! The whole solar system would be disrupted. I have no idea what "microwave warping of space" means.

I commend your son for his hypothesizing and you for your tutoring and encouragement of his curiosity.

QUESTION:
I understand that LED light won't refract a full spectrum of colors through a prism. Is this true? I want to combine a grouping of different wave length diodes to deliver a white light that will separate from indigo to red. Can this be done?

ANSWER:
Although a simple LED will not be monochromatic like a laser, it will tend to have a fairly narrow range of wavelengths emitted, not a full spectrum. The most common way to make a white LED is to take a blue or near-UV LED and coat it with a material called a phosphor. The phosphor will absorb some of the light from the LED and reemit that energy in a broader spectrum creating white light. The details of the spectrum depend on the phosphor used; varying the phosphor used, for example, can create the whites described as cool white or warm white or bright white; two of these are shown in the figure. I do not understand what "separate from indigo to red" means.

QUESTION:
As I understand it, electricity and magnetism are the same force; this is where "electromagnetism" comes from. I also understand that radio waves, light, and non-particle radiation like x-rays are all the same thing; waves in the magnetic field. However, they're carried by different particles: electrons for electricity, photons for magnetism (radio, light, etc). Furthermore, there is a significant difference between electrons and photons; electrons have mass, photons do not. Wouldn't this mean that electricity and magnetism are actually separate forces which are so tightly coupled (have a very strong influence on eachother), that it's just easier or simpler to think of them as the same force most of the time? One example of this coupling in my hypothesis is how electrons gain energy when a photon collides with and is absorbed by an electron, and the reverse, when an electron emits a photon and looses energy.

ANSWER:
You have some serious misconceptions here. First, the waves you refer to are not "waves in the magnetic field", they are waves which are composed of magnetic and electric fields—they are electromagnetic waves. Second, both electric and magnetic forces are "carried by" photons—electrons never play that role. Indeed, there is only one field, the electromagnetic field; for example, if you have a charged particle at rest you will only see an electric field, but if you give the charge some velocity you will now also see a magnetic field and you will find that the electric field is somewhat smaller which you could interpret as having "morphed" into the magnetic field.

QUESTION:
I want to reach 90% c in 30 days. How many Gs would I have to accelerate at to reach that velocity in the given time? (in a vacuum)

ANSWER:
You should read an earlier answer to understand the answer here. The graph here is from that earlier answer and shows (in red) v/c as a function of a₀t/c where a₀is the acceleration measured in the frame of the object which is accelerating and t is the time since the object was at rest. Reading off the graph, you can see that v/c =0.9 when a₀t/c=2. Now, t=30 days=2.6x10⁶ s, so a₀t/c=a₀(2.6x10⁶/3x10⁸)=8.64x10^-3a₀=2; solving, a₀=232 m/s²=23.6 Gs. I would not suggest having anybody aboard this object!

QUESTION:
Everything being equal,including speed and mass, Like a man walking up a slope, does the amount of energy expended on say a 10% inclined slope double with a 20% inclined slope?, if traveling at the same speed and everything else the same? In other words, is energy expended related arithmetically and constant according to slope of incline. and resistance, or is it proportional to the slope by some other factor, geometric increase.?

ANSWER:
In the figure the man of mass m is walking up a hill with incline θ with a constant speed of v. His potential energy is E=mgy where y is his height above the bottom of the hill and g=9.8 m/s² is the acceleration due to gravity. The rate at which his energy is changing is dE/dt=P=mg(dy/dt)=mgv_y=mgvsinθ; so this rate (called power) depends only on the vertical, not the horizontal component of his velocity. Now, you have made things difficult for me because you have not specified the angle, but rather the inclination which is actually the tangent of the angle (rise/run) times 100%. I will call this inclination the grade G=100tanθ. For example, let's take a man with mass 100 kg walking with a speed of 2 m/s up the 10% and 20% inclines you specify. The angles would be θ_10%=tan^-1(10/100)=5.71⁰ and θ_20%=tan^-1(20/100)=11.31⁰. So the power values are P_10%=100x9.8x2xsin(5.71)=195.0 Watts and P_20%=100x9.8x2xsin(11.31)=384.4 Watts. The ratio of these two is 1.97, close to, but not equal to, 2.0; for small angles the sine and tangent are approximately linear functions but as the angle (hence inclination or grade) increases this is no longer a good approximation. The graph shows the power for grades up to 200% (about 63.4⁰); as you can see, the power increases approximately linearly only up to about a grade of about 30% which is about 17⁰.

QUESTION:
A vehicle , unknown speed of travel, strikes a 14lb object and displaces it 29ft. Could you please tell me what the approximate speed of travel is and how you come to that conclusion. I ask because a vehicle deliberately veered to the opposite side of the street in order to run over my cat. The driver was successful in killing my sweet lad and I am trying to collect as many facts as I can before I request help in prosecuting the individual under Arizona's Animal cruelty laws.

ANSWER:
I am afraid that you cannot deduce the speed of the vehicle from the information you gave me. You need to know how elastic the collision was and what the trajectory of the cat was.

FOLLOWUP QUESTION:
Thank you for responding to my question but I am at a loss as to what is meant by "elastic". As far as trajectory my cat was launched a distance of 29 ft from point of impact with the next point where he died on the street...… no evidence, ie; of blood in between with there being blood evidence only at the point of impact and final resting location. If it would help I can shoot a video of the street and the location of where Liam was struck and where he landed in relation to the surroundings.

ANSWER:
Elastic refers to the energy lost in the collision which gives you information on how fast the cat was moving immediately after the collision. The extremes are a perfectly inelastic collision where the cat would be stuck to the vehicle after the collision and perfectly elastic which would have the cat moving with approximately twice the speed of the car after the collision. Clearly, neither of these would describe a car hitting a cat and I cannot imagine how you could approximate it.

When I say "trajectory" I refer to the path from collision point to the ground. The horizontal distance gone from the point of collision to the point of landing depends on the angle at which the cat started; if launched at 45⁰ relative to the horizontal he will go much farther than if launched at 20⁰. Again, unless you witnessed the accident, there is no way to know the angle.

QUESTION:
If you are traveling on a spaceship close to the speed of light (pick a number e.g. 0.95c), With or without constant acceleration whichever works for the question. Time will slow down relative to earth and you will be able to travel a greater distance than you would be able to without relativistic effects. You would also be able to return to earth with significant time passed relative to your experience. My question is, what would it feel like on the spaceship...I know the standard answer is you wouldn't feel anything, but logically, if you were traveling through galaxy after galaxy in a relativistic time of human life then intuition says everything would look like it whizzing past. Basically, what is the human experience of being able to travel interstellar distances?

ANSWER:
Of course everything is whizzing past compared with what it would look like if you were at rest relative to the earth. What you would also see is that the distance to an object you were moving toward is shorter by a factor of √(1-0.95²)=0.31; so if you went to a star 1 light year from earth, it would only take you 0.31/0.95=0.33 years to get there but the earth-bound clock would say it took you 1/0.95=1.05 years. Also where the stars appear to be relative to where they would be if you were not moving would be different; see an earlier answer. In the case where v/c=0.95, the graph shows how the angle at which you see something depends on where it would be if you were at rest; for example, a star actually at θ=120⁰ (30⁰ behind you) would be seen as being about θ'≈30⁰ in front of you!

QUESTION:
how much times earth to be compressed to become black hole?

ANSWER:
A black hole is a singularity, of zero size, so compression is infinite. If the earth were a black hole, its Schwartzchild radius, R=2GM/c², would be about 1 cm. R is the radius within which nothing can escape. So, if you could get the earth down to that size, it would then collapse the rest of the way to a black hole, I presume.

QUESTION:
If momentum is applied not on the center of mass on an object, what would its momentum be at the center of mass and is there an angular momentum at the point of impact?

ANSWER:
Linear momentum is not something which is applied, force is what is applied. And, what happens also depends on the details of how the force is applied. I will work out one simple example which will illustrate the principles involved when the force is applied in a simple collision. A mass m approaches the end of a thin uniform rod of mass M and length 2L with velocity v perpendicular to the rod; when it collides with the end of the rod it sticks. The center of mass of the rod is at its geometrical center a distance L from where m exerts the force. Linear momentum and angular momentum are both conserved but, as we shall see, energy is not conserved. After the collision the center of mass has a velocity u; conserving linear momentum, (M+m)u=mv or u=v[m/(M+m)]. Before the collision m brings in an angular momentum mvL. After the collision, the rod plus mass has angular momentum (I+mL²)ω where I=M(2L)²/12=ML²/3 is the moment of inertia of a thin rod about its center of mass. Conserving angular momentum I find ω[m+(M/3)]L²=mvL or ω=mv/{[m+(M/3)]L}.

Just for fun I put in some numbers to illustrate what would happen in a specific situation: m=1 kg, M=3 kg, L=1 m, v=4 m/s. With these I find: u=1 m/s, ω=2 radians/s=19.1 rpm. You can also calculate the energy before and after the collision: E_before=½mv²=8 J, E_after=½(M+m)u²+½([m+(M/3)]L²ω²=6 J. So, energy was not conserved in this collision, 2 Joules were lost.

CORRECTION:
I realized that an error had been made in my analysis. After the collision the center of mass is no longer at the center of the rod but displaced a distance x=mL/(M+m) toward where m is stuck. It is this center of mass which moves forward with speed u, and u is still v[m/(M+m)]. The rotation is about this center of mass but now the moment of inertia is different; the net (rod plus mass) moment of inertia is ML²/3+Mx²+m(L-x)², so the corrected value of the angular velocity is ω=mvL/[ML²/3+Mx²+m(L-x)²]. In the example I did, x=1/4 m, so ω=16/7 radians/s=21.8 rpm and E_after=46/7=6.57 J (1.43 J lost).

QUESTION:
I have been trying to figure this out, but having some trouble doing it: In general, let's say I had a vehicle that could accelerate me (according to my perception) at 1G for 1 year (again according to my perception), then turn around and decelerate (by my perception again) for another year to return home. How far would I have travelled and how old would my twin at home be when I got back?

ANSWER:
The reason you are having trouble is that relativistic kinematics involving acceleration, even if the acceleration in one frame is uniform, is not easy; the main reason is that, unlike Newtonian mechanics, acceleration in one frame is not the same as acceleration in a different frame. As a prelude to reading my answer, you should read an earlier answer and the earlier answers referred to in that answer. The graph here shows the results derived or cited in that earlier answer. The three graphs all show what is observed of your motion as seen by the twin on earth. The red curve is your velocity v relative to c as seen by the earth-bound twin, v/c=(gt/c)/√[1+(gt/c)²]; the blue curve is your acceleration a relative to g, a/g=[1+(gt/c)²]^-3/2; the black curve is your position expressed (approximately) in light years, gx/c²=√[1+(gt/c)²]-1. Now, since I have set this up as seen from earth, let me specify the time you travel as the time of your trip out as observed from earth: earth-bound twin observes you to arrive at you destination in two years. Then, as you can read off the graph, gx/c²=√[1+(2)²]-1=1.24 ly; the return trip will be the same length of time, so the earth-bound twin's clock will have advanced 4 years. Light would take 2.48 yr to travel that distance, so your clock will have advanced a time somewhere between 2.48 and 4 years. Your maximum speed would have been about v/c=(2)/√[1+(2)²]=0.89. We can get a rough estimate of your clock reading by approximating your average relative speed as 0.89/2=0.445; so you would have seen, on average, the distance contracted by √[1-(0.445)²]=0.9 and your time would have been 0.9x4=3.6 years.

QUESTION:
What happend to accelaration due to gravity if the earth stops rotating?

ANSWER:
Because of the centrifugal force, g appears to be somewhat smaller than it really is, but the amount is very tiny. This effect depends on your latitude and is zero at the poles and greatest at the equator. Also because of this effect, the acceleration is not directly toward the center of the earth except at the poles and equator.

QUESTION:
Hello, is the mass of a person on a spaceship (to provide a suitable upward force) important in designing a spaceship? My teacher says otherwise but upon researching I found out that additional fuel is required for every kg of payload that will be sent out into space.

ANSWER:
If the ship and all the fuel are much bigger than the mass of the person, then you could neglect the mass of the person in any calculations you know; in other words, the rocket launch would be, as close as you could hope to be measure, identical whether or not an astronaut was on board. However you are also right that the amount of fuel you need depends on the payload you want to deliver, and that would include everything which is not fuel. You might find the an earlier answer illuminating.

QUESTION:
How exactly do electrons in electric discharges generate radio waves? Whenever someone uses an AM radio while there are electric sparks like lightning strikes nearby the radio waves they emit can be detected.

ANSWER:
Any time an electric charge is accelerated it emits electromagnetic radiation. There are many electrons in the spark which accelerate under the influence of the electric fields they experience.

QUESTION:
I just read about the kilogram being defined by measuring the current of an elector-magnet to balance a scale. How can various electromagnets have exactly the same force to current ratio? Would variations in the purity of the conductor, variations in the exact coil shape, etc. cause variations in this ratio? What am I missing about this new designation?

ANSWER:
You cannot use any old electromagnet. There is an exquisitely accurate balance called the Kibble balance which is used. Also see Wikipedia for more detail

QUESTION:
Is the following statement correct? Net radiated energy is made up of emitted and absorbed energy.

I stated that it was incorrect based on the fact that "made up of" usually means putting two or more things together (aka addition). Net radiated energy is the difference between emitted energy and absorbed energy. I have yet to see my exam paper but I believe that the mark I lost was due to this question on basis that “made up of” means that it is a result of a mathematical relation between the two values and not necessarily addition (in this case subtraction). What is the correct answer in this case and why?

ANSWER:
I do not believe that there is a hard and fast definition of what "net radiated energy" is. It is a matter of semantics. However, radiated energy usually means energy emitted and absorbed energy means energy absorbed; therefore, I would say that this is an incorrect statement because net radiated energy would mean the sum of all radiated energy, absorbed energy having nothing to do with it. Your reason is quite shaky because "made up of" is even more semantically ambiguous than "net radiated"; it would simply imply that "net" means total energy flux.

QUESTION:
How is momentum conserved for single or double slit diffraction? in other words:I have a photon gun that I can dial the intensity down to fire one photon at a time. I fire my photon gun, the gun recoils in the -X direction, the photon flies off in the +X direction. The photon then passes thru two closely spaced slits (double slit experiment)instead of appearing on the screen directly behind the slits, the photon ends up to the side (after many, many photons the interference pattern appears) however, for the single event of a single photon, how is momentum conserved? dos the single event photon exchange momentum with the walls of the slit? Since the photon ended up to the side, it was no longer travelling in the purely X direction, therefore shouldn't the photon have exchanged momentum with the slit somehow? If the photon interacts with the slit, that counts as “which way” information, and the interference pattern vanishes. If the interference pattern remains, then how do we account for the photons change in momentum? This is for a single event, one photon, not expectation value of many, many photons.

ANSWER:
You are forgetting that the photon goes through both slits, is being forced (if you like) to assume its wave-like identity. It behaves like a wave until something makes it act like a particle, so it is a wave spreading out in both directions, so its net momentum in the direction parallel to the screen is zero. When the screen is encountered it is a "measurement" which collapses the wave function at some point.

QUESTION:
I can imagine 2 similar objects rotating around a sphere at the same velocity but at different orbits. One object would rotate around the sphere faster than the other, right? What is puzzling to me is that if I think of both objects going now in a straight line one could not say that one object was going faster than the other. Can someone help me through this? Is there a subtlety here?

ANSWER:
You want to say same speed, not same velocity, but I get the idea. Say one orbit is twice the radius as the other. Then the smaller orbit has a circumference half that of the larger. So each time the object in the larger orbit goes around once, the one in the smaller orbit goes around twice. But they still have the same speed. But the angular speed of the object in the smaller orbit is twice that of the object in the larger orbit even though their linear speeds are the same. You cannot do this with planets or moons, though, because the speed of a satellite in a circular orbit depends on the radius.

QUESTION:
If space is curved, doesn't it need another dimension to curve into? If a 2d object is curved it curves into the 3rd dimension. So if 3d space is also curved it must curve into 4d.

ANSWER:
This is really mathematics, not physics; I am not a mathematician. But the definition of an N-dimensional space is, I believe, that N quantities are required to specify the location of a point in that space. Curvature of a space is defined as a space in which the geometry is not Euclidean. The physical existence of an N-dimensional space does not imply the physical existence of an N+1 dimensional space. You cannot make a generalization that curvature is necessarily "curving into" a higher dimension. For example, a planar sheet can be distorted without leaving the 2D. You also might "curve into" a higher dimension than N+1; for example, a coil spring is a one dimensional space which exists in a 3D space. You are trying to make generalizations based on the only spatial dimensions which we know to exist; but higher dimensions are defined by certain rules which in no way prove that they exist physically.

While researching this question I came upon a very interesting little essay about general relativity, curvature of spacetime, gravity. Many nonscientists (and scientists as well) are familiar with and accept as gospel that general relativity demonstrates that the distortion of spacetime by the presence of mass is what gravity is; the "cartoon" which illustrates this is the "bowling ball on a trampoline" example in which a mass causes the distortion of the two-dimensional space defined by the trampoline surface. The implication is that gravity is just geometry. On the other hand, many theorists are struggling to find a successful theory of quantum gravity and to do that one must treat gravity as a force. The take-away from this essay, though, is that curved spacetime is not a unique representation of what the mathematics which are the framework of the very successful theory of general relativity mean. The author, Rodney Brooks, notes that "…you can view gravity as a force field that, like the other force fields in QFT [quantum field theory], exists in three-dimensional space and evolves in time according to the field equations." He also goes so far as to say "…shame on those who try to foist and force the four-dimensional concept onto the public as essential to the understanding of relativity theory."

QUESTION:
A rear wheel drive car accelerates quickly from rest. The driver observes that the car noses up. Why does it do that? Would a front wheel drive car behave differently?

ANSWER:
The car with rear-wheel drive has an acceleration a and a mass M. The forces on the car are its own weight W (W=Mg) acting at the center of mass a distance d from the real axle; the normal and frictional forces on the rear wheels, N₁ and f₁; and the normal and frictional forces on the front wheels, N₂ and f₂; the radius of the wheels is R and the distance between front and rear axels is D. I will ignore the rotational motion of the wheels, that is, I will approximate their masses as zero. If the front wheels are not lifting up from the ground, the sum of torques about the rear axel is (f₁-f₂)R+N₂D-Mgd=0; Newton's first law for the horizontal and vertical directions are (f₁-f₂)=Ma and N₁+N_₂-Mg=0. From these equations it is easy to show that N₂=M(gd-aR)/D. This is interesting because it shows that N₂=0 if a=gd/R; for larger accelerations, the front wheels will lift off the ground. (Note that as N₂ approaches zero, f₂ also approaches zero.) I have ignored the suspension of a real car, so this lift will not be abrupt; it will happen gradually as the front springs decompress and the rear springs compress; so overall the nose goes up but the rear end goes down.

If it is a front-wheel drive, the directions of the frictional forces reverse. The corresponding torque equation (this time about the front axle) is (f₂-f₁)R-N₁D+Mg(D-d)=0; Newton's first law for the horizontal and vertical directions are (f₂-f₁)=Ma and N₁+N_₂-Mg=0. From these equations it is easy to show that N₁=M(aR+g(D-d))/D. As a increases in magnitude, N₁ gets bigger and bigger until N₁=Mg which necessarily means that N₂ becomes zero. It is easy to show that this happens when a=gd/R, exactly the same result as for the rear wheel drive.

Finally, it might appear that there is no difference between front- and rear-wheel drive. That is not the case in the real world at all. Keep in mind that the frictional forces are static friction and the larger the corresponding normal force is, the more friction you can get. Since in the rear-wheel situation the normal force gets bigger, the amount of friction you can get without the wheels slipping can keep increasing so that eventually (with a powerful enough engine) you could lift the whole front end far off the ground. However, when you have front-wheel drive, the normal force on the front wheels gets smaller as you increase a, so most likely the wheels will slip well before N₂=0 when f₂ must be zero. For very large accelerations, like required for drag racers, rear-wheel drive is imperative.

QUESTION:
Gravity is what makes objects orbit around other objects,and gravity is a reflection of an object's mass.So why doesn' the mass of the objects appear in kepler's third law??

ANSWER:
For the same reason why all objects have the same acceleration in the gravitational field, i.e. all objects near the surface of the earth have an acceleration of 9.8 m/s². The acceleration (which is what describes the orbit) is F/m but the force is MmG/r², so the acceleration is MG/r².

QUESTION:
If kinematics can give you data on a dropped object, would it be valid to use kinematics to calculate the acceleration experienced by the object upon impact with the ground? (the time from when it hits the ground to when it actually stops)

ANSWER:
When the object hits the ground until its velocity reaches zero is some time, call it Δt. Its momentum drops from mv (where m is the mass and v is the speed when it hits) to zero because an impulse I was delivered to it by the ground. The impulse may be written as I=F_avgΔt. where F_avg is the average force. Therefore you can find the average acceleration a_avg during the collision, but not the acceleration as a function of time: a_avg=F_avg/m=I/(mΔt)=v/Δt. So you cannot get detailed kinematics about the collision without measuring how the speed decreases with time.

QUESTION:
We know that air density decreases as you ascend into space, but what about looking out to the horizon? Is there a atmospheric lensing that occurs? Is there a formula for its calculation?

ANSWER:
Certainly there is refraction due to this density profile of the air. There is no simple formula because the actual density at any point is dependent on the pressure and temperature of the air as well as altitude, and these vary with time. One could calculate based on average density profiles, but I would suppose it is a numerical calculation, not an analytic formula. There is a nice description of this phenomenon here.

QUESTION:
I've been interested in physics for years but have never managed to get my head around the concept of space-time. I understand time and i understand space but somehow i could never get this. However today i think it finally clicked when i wasn't even trying. Can you tell me if my understanding of it is correct or not? I thought of space-time as like if i were on a roller-coaster or train tracks. When i am on the long flats, i travel slower, and when i am on the declines, i travel faster. Is space-time like this analogy? In that, when you are on Earth for example, you experience time moving faster than if you were floating in space millions of miles away. This is because the presence of Earth "warps" space time, much like a bowling ball on a trampoline. This "dent" in space-time causes a gravitational pull, pulling time into it, much like if i were rolling down a hill. I know that might sound very convoluted but it's the idea i have in my head and it has bothered me for YEARS, but i think i am on to an understanding here.

ANSWER:
Sometimes we make things more difficult than they need to be. The main takeaway from the theory of relativity is that time and space are not separate and universal things. In Newtonian times, it was assumed that any clock in the universe would run at exactly the same rate; now we know that if a clock moves with respect to us that it runs more slowly than ours—doesn't appear to run slower, it runs slower. Similarly, space depends on the observer; a meter stick moving by you is actually shorter than 1 m. What these facts about space and time tell us is that time and they are not different things, they are really entangled with each other. You cannot talk about one without talking about the other. They are two pieces of the same thing. So the term space-time is simply a way of acknowledging that. I do not find your roller coaster analogy very helpful. The bowling ball on a trampoline is a classic but keep in mind that it is really a cartoon and not rigorous. Generally trying to picture something in four dimensions futile!

QUESTION:
What does a wind speed (a gust is okay)in miles per hour, need to be to launch a baseball at a speed of 80 miles per hour into an object like a window. Given: Ball is 60 feet 6 inches from the window that it hits traveling at 80 mph Baseball is 9.25 inches in circumference and 2.94 inches in diameter and weighs 5.25 ounces My name is Karin and I'm testing to see if I use a baseball pitching machine from 60 feet 6 inches from a window protected with a protective panel, and set the machine to pitch at 80 mph, what would the hurricane wind speed need to be to get that baseball to fly into the window at 80 mph. I hope this is clear enough.

ANSWER:
If I understand your question, you are essentially asking what wind speed would be required for a baseball to reach a speed, starting from rest, of 80 mph in a distance of 60.5 ft due to the force of the wind on the ball. This is a problem which is very tedious to do in full analytic detail and not very illuminating. Also, all problems involving air drag are approximate, so it would be appropriate for me to find an approximate way to solve this problem. I am going to work in SI units so I will rephrase your question: what wind speed would be required for a baseball to reach a speed, starting from rest, of 35.8 m/s in a distance of 18.4 m due to the force of the wind on the ball? In SI units you may approximate the force at sea level which the wind exerts on the ball as F=¼A(u-v)² where u is the speed of the wind, v is the speed of the ball, and A is the cross sectional area of the ball. Applying Newton's second law, ¼A(u-v)²=ma=mdv/dt . This differential equation may be shown to have the solution t=cv/[u(u-v)] where c=4m/A and t is the time it takes the ball to reach the speed v if the wind speed is u. I find that c=137 s²/m. Note that after a very long time the speed will be constant at the wind speed; we are only interested in the time it takes to get to 35.8 m/s (80 mph). I have plotted t for three values of u, 70, 80, and 90 m/s.

Now comes the hard part. We are interested in finding the value of u for which v will be 35.8 m/s when the ball has gone a distance x=18.4 m. This means that we need to write the equation for t as t=c{dx/dt}/[u(u-{dx/dt})] and solve that differential equation for t as a function of x. That is actually doable, but the solution is so messy that I looked for a simpler way to do it. This took some trial and error, but I got a pretty good estimate, I believe. Look at the black line in the graph; it is not a straight line, but it is pretty close, so I am going to approximate the acceleration as being constant over this time interval of 1 s, a≈(35.8 m/s)/(1 s)=35.8 m/s². Now, an object with constant acceleration starting from rest goes a distance d=½at² in a time t, so if u=90 m/s=201 mph the ball will have a speed of 35.8 m/s when d=35.8/2=17.9 m=59 ft. As far as I am concerned, I have adequately solved this problem—it takes a wind of approximately 200 mph to accelerate a baseball from rest to the speed of 80 mph in a distance of about 60 ft.

QUESTION:
Why is light speed the fastest speed and why would people see a train going backwards if it is traveling at a velocity greater than light speed?

ANSWER:
I will not answer the second question because the site groundrules state that I will not answer questions of this sort. Your first question is equivalent to the question "why can nothing with mass travel as fast or faster than light?" which is included in the FAQ page.