An elastic collision takes place between two smooth, rubber balls of same radius as shown. Initially, one ball is at rest and the other is moving with velocity u. At maximum compression :
I. Ratio of potential to initial kinetic energy of the system is `(cos^2 theta)/(2)`
II. Ball B is moving along direction a
III. Ball A is moving along direction d
Evaluate the above statements and choose the correct option from the following:

Options

A. Statements I, II are true and III is false
B. Statements I, III are false and II is true
C. All statements are true
D. All statements are false

Answer

A

Explanation

At the time of maximum compression velocities along common normal becomes equal.
Let it be V.

`implies " "` along common normal: ` m u cos theta = 2 mv implies v = ( u cos theta)/(2)`
Total KE at this moment
`=1/2 {((u cos theta)/(2))^(2) + ( u sin theta)^2 } + 1/2 m ((u cos theta)/(2))^(2)`
`= 1/2 m u^2 {(cos^2 theta)/(2) + sin^2 theta} = 1/2 m u^2 {1 - (cos^2 theta)/(2)}`
`implies PE = 1/2 m u^2 - 1/2 m u^2 { 1 - (cos^2)/(2)} = 1/2 m u^2 (cos^2 theta)/(2) implies` ratio of PE to intial KE = `(cos^2 theta)/(2)`.
Absolute velocity vector of A lies between common tangent and common normal and velocity vector of B lies along common normal.