Question

ABC is a frictionless circular track of radius R. A particle of mass m kg is released from point P (OP = r/2). After collision with the track. Particle moves along the track, than coefficient of restitution is:

A) 0.5
B) 0.3
C) 0.2
D) None

Answer 1

Hint:Coefficient of restitution (e): coefficient of restitution can be defined as the ratio of relative velocity between two objects when they separate to the relative velocity when they approach each other. For this question try to calculate the approach velocity and separating velocity normal to the colliding point and then take their ration and you will get the answer.

Complete step by step answer:
The degree to which a collision is elastic or inelastic is quantified by the coefficient of restitution, a value that generally ranges between zero and one. A perfectly elastic collision has a coefficient of restitution of one; a perfectly-inelastic collision has a coefficient of restitution of zero.
Coefficient of restitution: \[e = \dfrac{{|{v_1} - {v_2}|}}{{|{u_1} - {u_2}|}}\]
Where
\[|{u_1} - {u_2}|\] represents relative approaching velocity
\[|{v_1} - {v_2}|\] represents relative separation velocity
Now let us try to calculate the approaching velocity and separating velocity
As it is said in the question that when the ball hits the track it moves along the surface that is after hitting the ground the velocity of the ball normal to the track where it falls is zero because it is moving perpendicularly.
Self made image :

\[|{v_1} - {v_2}| = 0\]
\[|{u_1} - {u_2}| = V\]
Where \[V\] has some value.
Now putting these two values in given formula we get:
\[e = \dfrac{0}{V} = 0\]
Thus, we can say the coefficient of restitution is zero, that is collision is perfectly elastic.

Hence, option D is correct.

Note:Remember these points while solving such questions:
-Collision is instantaneous interaction of two bodies when they collide with each other which may result in change of their momentum due to internal forces acting between them during this.
-Collisions can either be inelastic, meaning they conserve momentum but not kinetic energy or elastic, meaning they conserve both momentum and kinetic energy.
-When dealing with an incident body that is nearly parallel to a surface, it is sometimes more useful to refer to the angle between the body and the surface, rather than that between the body and the surface normal.