Question

In an experiment, two protons are shot directly toward each other, each moving at half the speed of light relative to the laboratory.

a What speed does one proton measure for the other proton?

b What would be the answer to part (a) if we used only nonrelativistic Newtonian mechanics?

c What is the kinetic energy of each proton as measured by an observer at rest in the laboratory?

d What is the kinetic energy of each proton as measured by an observer riding along with one of the protons?

e what would be the answers to part (c) if we used only nonrelativistic Newtonian mechanics?

f What would be the answers to part (d) if we used only nonrelativistic Newtonian mechanics?

Answer 1

Two protons have speed

Now consider one of the protons as S frame and other one as S' frame of S' is moving with speed c/2 and S is at rest so velocity measured by one proton

Part b

In Newtonian mechanics

Part C

which can also be written as

So,

Part d

As the observer is riding with one of the protons the speed is v = 4c/5, so

Part e

in nonrelativistic mechanics the kinetic energy is

Since v = c/2

part f

In the case of d , the relative velocity is v = c , so,