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174. Problem 21.54 (RHK) (a) Suppose we have a particle accelerated from rest by
the action of a force F. Assuming that Newton’s second law for a particle,
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, is
valid in relativity. We have to show that the final kinetic energy K can be
written, using the work-energy theorem, as
.
(b) By substituting the expression for relativistic momentum and carrying out
the integration, we have to show that 
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Solution: Click For PDF Version (a) For simplicity we will consider one dimensional motion. According to Newton’s second law
where p is the momentum of the particle at time t. When the
particle moves by distance
(b) Relativistic momentum of a particle of mass m and velocity v is given by
Therefore,
We, therefore, have
Let us make the substitution in the integration variable
This gives
And, we have
The energy of a particle of mass m moving with speed v will therefore be
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in the
time interval
, then
according to the work-energy theorem the change in the kinetic energy of the
particle will be
