Reference: Carroll, Bradley W. & Ostlie, Dale A. (2007), An Introduction to Modern Astrophysics, 2nd Edition; Pearson Education – Chapter 4, Problem 4.15.
Under a constant force , an object undergoes hyperbolic motion. In one dimension for a constant force we have
where is a constant of integration. If the object starts at at rest (in the lab frame), then , and
which can be solved for the velocity to give
We can also get this formula by integrating the expression for the acceleration
If is constant and acts on an object initially at rest, then and
To find we integrate:
If at , the constant of integration is and we get 3 again.
From 3 we can get the inverse function
Thus as , so the object never quite reaches the speed of light.
Because of the velocity-dependent term in 5, the acceleration due to a constant force is not constant, but rather decreases as increases. If we start with , then the times required to reach various speeds are found from