Question

Calculate the velocities of the two balls after a collision for following special cases, when -
(a) both the balls are of equal masses.
(b) the balls are of unequal masses and the mass of one ball is very-very large as compared to the other.​

Answer 1

Explanation:

Let m1 and m2 denote the masses of the balls, where m1>m2, and v denote their initial speed. Let the positive direction of motion be the direction in which ball 1 is initially traveling. Define both balls as the system. The linear momentum of the system before the collision is m1*v-m2*v=(m1–m2)*v. Note that the sign of the linear momentum is positive. The closing speed of the two balls is 2v. Since the collision is elastic, the separation speed of the balls after the collision equals the closing speed before the collision. Now determine which ball becomes stationary. If ball 1 remains stationary after the collision, ball 2 must have a nonnegative velocity after the collision, rendering certain a positive final linear momentum of the system. If ball 2 remains stationary after the collision, ball 1 must have a non-positive velocity after the collision, rendering certain a negative final linear momentum of the system. Since linear momentum of the system is conserved, it must be the case that ball 1 remains stationary after the collision while ball 2 reverses its direction of motion. Applying the conservation of linear momentum, where the final speed of ball 2 is V:

(m1-m2)*v=m2*V

Since ball 1 is stationary, the speed of ball 2 equals the speed of separation, which is equal to 2v.

(m1-m2)*v=m2*2v

m1-m2=2*m2

m1=3*m2

Therefore, the heavier ball is 3 times as massive as the lighter ball.