Question

Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with aspeed V. If the collision is elastic, which of the following (Fig. 6.14) is a possible result after collision ?

Answer 1

For the case of an elastic collision, the total kinetic energy should remain constant before and after the collision.

Initially, the velocity of the colliding ball bearing is V and the other two ball bearing of same mass m is zero.

Hence, the expression for kinetic energy before collision is given as,

K i = 1 2 m ( V ) 2 + 1 2 ( 2m )( 0 ) = 1 2 m ( V ) 2

(i).

The following figure shows the collision of the ball bearing for case (i).

Here, the final velocity of ball bearing 1 is 0 while that of 2 and 3 is V 2

The expression for total kinetic energy after collision K f1 in this case is given as,

K f1 = 1 2 ×m×( 0 )+ 1 2 ( 2m ) ( V 2 ) 2 = m ( V ) 2 4

In the above case,

K i ≠ K f1

The kinetic energy after the collision and before the collision is not same. Hence, it is not a case of elastic collision.

(ii).

The following figure shows the collision of the ball bearing for case (ii).

Here, the final velocity of ball bearing 1 and 2 is 0 while that of 3 is V.

The expression for total kinetic energy K f2 after the collision is,

K f2 = 1 2 ( 2m )×( 0 )+ 1 2 m ( V ) 2 = 1 2 m ( V ) 2

In the above case,

K i = K f2

The kinetic energy after the collision and before the collision is same. Hence, it is a case of elastic collision.

(iii).

The following figure shows the collision of the ball bearing for case (iii).

Here, the final velocity of ball bearing 1, 2 3 is V 3 .

The expression for total kinetic energy after collision K f3 in this case is,

K f3 = 1 2 ( 3m ) ( V 3 ) 2 = 1 6 m ( V ) 2

In the above case,

K i ≠ K f3

The kinetic energy after the collision and before the collision is not same. Hence, it is not a case of elastic collision.

Therefore, only (ii) shows an elastic collision.