A particle strikes a smooth floor at an angle 37 degree with the normal and rebound at an angle 53 degree with the normal . The coefficient of restitution is

Let the ball strikes the plate with speed v.

Suppose v' is the speed after the collision and θ' is the angle of reflection. 

Parallel component of velocity of the ball remains constant as there is no force parallel to the surface. 

v'sinθ' = vsinθ  ...(i)

For component normal to the floor,

velocity of the separation = v'cosθ'

velocity of approach = vcosθ

Therefore, e = (v'cosθ')/(vcosθ)  

v'cosθ'=evcosθ  ...(ii)

From (i) and (ii) we get,

tanθ' = tanθ/e

But 53 and 37 are complementary angles

Hence

e = 1

Regards

Arun (askIITians forum expert)