Final answer:
The displacement z of a particle with rest mass m0 subject to a constant force m0g along the z-axis, accounting for relativistic effects, involves integrating the work-energy theorem's relativistic expression for kinetic energy and comparing it with the classical displacement z = 1/2 gt².
Step-by-step explanation:
The problem given involves finding the displacement z of a particle with rest mass m0 under the influence of a constant force along the z-axis, considering relativistic effects. The comparison with the classical result implies a consideration of the relativistic factor y in kinetic energy expressions.
We recall that the relativistic kinetic energy is more complex than the classical expression K = 1/2 mu², because as the velocity u of the particle approaches the speed of light c, the kinetic energy diverges due to the increase in y.
The task involves using the work-energy theorem in a relativistic context, where the work done by the force m0g produces a change in the relativistic kinetic energy. As such, relativistic mechanics dictates that energy and momentum are dependent on the relativistic factor y, which is a function of velocity.
Therefore, to expand the displacement z as a power series in time t, one would start by expressing the relativistic momentum and kinetic energy, then integrate to find the time-dependent expressions for velocity and displacement.
Comparing with the classical result, where z is simply obtained by z = 1/2 gt² (ignoring air resistance), shows how relativistic effects modify the simple parabolic trajectory of a particle in a gravitational field. The relativistic displacement would involve a series with higher-order terms in time, accounting for the varying mass with velocity.