An Introduction to Modern Astrophysics · Problem 4.15

The Interaction of Light and Matter — Carroll & Ostlie

Physics — Astrophysics · relativistic-acceleration

Problem

Suppose a constant force of magnitude

F

acts on a particle of mass

m

initially at rest. (a) Integrate the formula for the acceleration found in Problem 4.14 to show that the speed of the particle after time

t

is given by

\frac{v}{c} = \frac{(F/m)t}{\sqrt{(F/m)^2 t^2 + c^2}}.

(b) Rearrange this equation to express the time

t

as a function of

v/c

. If the particle's initial acceleration at time

t = 0

a = g = 9.80

m s

^{-2}

, how much time is required for the particle to reach a speed of

v/c = 0.9

v/c = 0.99

v/c = 0.999

v/c = 0.9999

v/c = 1