Question

A body falling freelynfromfa given height H Hits an inclined plane in its path at a height h. As a result of this impact the direction of the velocity of the body becomes horizontal. For what value of h/H, the body will take the maximum time to reach the ground.

Answer 1

Suppose, the collision of the body with the plane is elastic and there is no loss in kinetic energy. Now after falling from height H it strikes with an inclined plane at a point 'h' above the ground. Thus distance travelled by the body before impact = (H - h)

Initial velocity of the body = 0

So time taken is given by, S = 0 + 1/2 g t2

=> t = [2(H-h)/g]1/2

Now after striking the body has horizontal component of velocity only.

Thus it falls freely under the gravity with initial vertical component of velocity = 0

Again, using S = 1/2 g t2

=> h = 1/2 g t2

=> t = [2h/g]1/2

Thus total time taken by body to reach the ground = T = [2(H-h)/g]1/2 + [2h/g]1/2

Now for extreme conditions,

dT/dh = 0

=> dT/dh = 0.5 [ 2(H-h)/g ]-1/2 × (-2/g) + 0.5 [2h/g]05 × (2/g)

=> dT/dh = - [ 2(H-h)/g ]-1/2 /g + [2h/g]-1/2 /g

=> [ 2(H-h)/g ]-1/2 /g + [2h/g]-1/2 /g = 0

=> [2(H-h)/g ] = [2h/g]

=> 2H/g = 4h/g

=> h = H/2

=> h/H = 1/2