Pushing on a mass A mass m starts at rest Starting at t 0 y

Pushing on a mass A mass m starts at rest. Starting at t = 0, you push on it with a constant force F. How much t does it take for the mass to move a distance x? (F, t, and x here are measured in the lab frame.) Check that your result is reasonable when x rightarrow 0. After a very long time, m\'s speed will approach c. It turns out that it approaches c sufficiently fast so that after a very long time, m will remain (approximately) a constant distance (as measured in the lab frame) behind a photon that was emitted also at t = 0 from the starting position of m. What is the distance?

Solution

(a) If a is the acceleration

F=ma

a = F/m

x = ut+(1/2)at² [u=0]

x = (1/2)at²

t = (2ax)^1/2 = (2Fx/m)^1/2 (1)

(b) when v = c

v = u + at

Using eq (1)

c = 0 + (F/m)(2Fx/m)^1/2   

c² = (F/m)²(2Fx/m) = 2F³x/m³

x = m³c² /2F³