Classical Mechanics (Kibble & Berkshire) · Problem 4.11

Linear oscillations — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · straight-line-motion-inverse-cube-gravity

Problem

A particle

P

of mass

m

is attracted towards a fixed origin

O

by a force of magnitude

m\gamma/r^3

, where

r

is the distance of

P

from

O

and

\gamma

is a positive constant. [It's gravity Jim, but not as we know it.] Initially,

P

is at a distance

a

from

O

, and is projected with speed

u

directly away from

O

. Show that

P

will escape to infinity if

u^2 > \gamma/a^2

. For the case in which

u^2 = \gamma/(2a^2)

, show that the maximum distance from

O

achieved by

P

in the subsequent motion is

\sqrt{2}a

, and find the time taken to reach this distance.