Classical Dynamics of Particles and Systems · Problem 2-50

Oscillations — Marion & Thornton

Physics — Mechanics · relativistic-particle-constant-force

Problem

According to special relativity, a particle of rest mass

m_0

accelerated in one dimension by a force

F

obeys the equation of motion

dp/dt = F

. Here

p = m_0v/(1 - v^2/c^2)^{1/2}

is the relativistic momentum, which reduces to

m_0v

for

v^2/c^2 \ll 1

. (a) For the case of constant

F

and initial conditions

x(0) = 0 = v(0)

, find

x(t)

and

v(t)

. (b) Sketch your result for

v(t)

. (c) Suppose that

F/m_0 = 10

m/s

^2

(

\approx g

on Earth). How much time is required for the particle to reach half the speed of light and of 99\% the speed of light?