First of all, it is not correct that pc is the kinetic energy of a particle. Actually, kinetic energy T is (by definition) the total energy E minus the rest mass energy mc2. More about kinetic energy below, but I will solve the first part of the problem assuming that I want a 1% error if I make the approximation that . If you want to do it over assuming that , be my guest! I think it will be more algebraically involved and less illuminating. (Incidentally, whenever I write m I mean rest mass.)
I have the feeling that maybe you and I have addressed the root of your problem here before, but I may be wrong. The key to being able to do this problem is to be able to do a binomial expansion; this is often the key to doing problems requiring approximations. The appropriate expansion is . This approximation works if and does not depend on n being an integer. Now, to your problem:
.
So, you see, the error you make in approximating is . Now, you are told that where . Now, let’s evaluate .
If you now do the algebra,
So, you see, the answer you gave me (0.9995) was wrong. Or maybe I do not understand what is meant by a 1% error, but if you take my answer and compute E you will get 1.01pc.
While we are at it, let’s address your second question about kinetic energy T:
This, as you see, is a little trickier because you must retain three rather than two terms of the expansion, otherwise you would simply get exactly . I presume you want to know the maximum v where you make no more than a 1% error, i.e.
.
Solving,