Classical Mechanics (Kibble & Berkshire) · Problem 2.10

Newton's laws of motion and the law of gravitation — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · pursuit-problems

Problem

A pursuit problem: Daniel and the Lion The luckless Daniel (

D

) is thrown into a circular arena of radius

a

containing a lion (

L

). Initially the lion is at the centre

O

of the arena while Daniel is at the perimeter. Daniel's strategy is to run with his maximum speed

u

around the perimeter. The lion responds by running at its maximum speed

U

in such a way that it remains on the (moving) radius

OD

. Show that

r

, the distance of

L

from

O

, satisfies the differential equation

\dot{r}^2 = \frac{u^2}{a^2}\left(\frac{U^2 a^2}{u^2} - r^2\right).

Find

r

as a function of

t

. If

U \geq u

, show that Daniel will be caught, and find how long this will take. Show that the path taken by the lion is an arc of a circle. For the special case in which

U = u

, sketch the path taken by the lion and find the point of capture.

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