JEE Main · 2024 · 31 Jan 2024 · Shift 1 · Kinematics

A body starts falling freely from height H hits an inclined plane in its path at height

h

. As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. The value of

\frac{H}{h}

for which the body will take the maximum time to reach the ground is ____.

Answer: 2

Solution

Let the body fall through

(H - h)

before hitting the incline (time

t_{1} = 2 (H - h) / g

); since the impact turns the velocity horizontal, it then falls the remaining height

h

as a horizontal projectile (time

t_{2} = 2 h / g

).

Total time:

T = \frac{2 ( H - h )}{g} + \frac{2 h}{g}

Maximising

T

with respect to

h

(for fixed

H

\frac{d T}{d h} = - \frac{1}{2 g ( H - h )} + \frac{1}{2 g h} = 0 ⟹ H - h = h ⟹ H = 2 h

\frac{H}{h} = 2

JEE Main · 2024 · 31 Jan 2024 · Shift 1 · Integer answer

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