Elastic collision on an inclined plane

Classical Mechanics Level 3

A body released from a height H H above the ground hits elastically an inclined plane at a point P P . After the impact, the body starts moving horizontally and hits the ground.

Find the height at which point P P should be situated so as to have the total time of travel maximum.

H 2H H/2 H/3 H/6 H/4

Ritwik Jain by ritwik jain , 18, India

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1 solution

Kelvin Hong
Jun 7, 2017

The inclined plane should has 4 5 ∘ 45^\circ with horizontal so the body could go horizontal after the moment it impacts.

At both t 1 t_1 and t 2 t_2 state, the body start from zero vertical velocity .

so we have

H − P = 1 2 g t 1 2 H-P=\frac{1}{2}gt_1^2

P = 1 2 g t 2 2 P=\frac{1}{2}gt_2^2

We want t 1 + t 2 t_1+t_2 be maximized, so

( t 1 2 + t 2 2 ) ( 1 + 1 ) ≥ ( t 1 + t 2 ) 2 (t_1^2+t_2^2)(1+1) \geq (t_1+t_2)^2

by Cauchy Inequality.

We see that

H = 1 2 g ( t 1 2 + t 2 2 ) H=\frac{1}{2}g(t_1^2+t_2^2)

t 1 2 + t 2 2 = 2 H g t_1^2+t_2^2=\frac{2H}{g}

It leads to

t 1 + t 2 ≤ 2 H g t_1+t_2 \leq 2\sqrt\frac{H}{g}

Equality holds at t 1 = t 2 t_1=t_2 by definition.

So

H − P = P H-P=P

P = 1 2 H P=\boxed{\frac{1}{2}H}

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