Classical Mechanics (Kibble & Berkshire) · Problem 4.22

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

Physics — Mechanics · projectile-clearing-barrier

Problem

It is required to project a body from a point on level ground in such a way as to clear a thin vertical barrier of height

h

placed at distance

a

from the point of projection. Show that the body will just skim the top of the barrier if

\left(\frac{ga^2}{2u^2}\right)\tan^2\alpha - a\tan\alpha + \left(\frac{ga^2}{2u^2} + h\right) = 0,

where

u

is the speed of projection and

\alpha

is the angle of projection above the horizontal. Deduce that, if the above trajectory is to exist for some

\alpha

, then

u

must satisfy

u^4 - 2ghu^2 - g^2a^2 \geq 0.

Find the least value of

u

that satisfies this inequality. For the special case in which

a = \sqrt{3}h

, show that the minimum projection speed necessary to clear the barrier is

(3gh)^{\frac{1}{2}}

, and find the projection angle that must be used.