Question

Imagine two billiard balls on a pool table. Ball A has a mass of 7 kilograms and ball B has a mass of 2 kilograms. The initial velocity of the ball A is 6 meters per second to the right, and the initial velocity of the ball B is 12 meters per second to the left.

1. When the two balls hit each other, what will happen if it is a perfectly
inelastic collision?

2. Compare the final velocity of the balls.

3. What can you say about the total momentum before and after the collision?

Answer 1

1) In a perfectly inelastic collision, the two balls stick together after the collision. In this type of collision, the total kinetic energy of the system is not conserved, while the total momentum is conserved.
If we call
`v_f` the final velocity of the two balls that stick together, the conservation of the total momentum before and after the collision can be written as

`m_a v_(Ai) + m_b v_(Bi) = (m_A+m_B)v_f` (1)
where

`m_A=7 kg` is the mass of ball A

`m_B=2 kg` is the mass of ball B

`v_(Ai)=6 m/s` is the initial velocity of ball A

`v_(Bi)=-12 m/s` is the initial velocity of ball B (taken with a negative sign, since it goes in the opposite direction of ball A)

If we solve (1) to find
`v_f`, we find that the final velocity of the balls is

`v_f= (m_Av_(Ai)+m_Bv_(Bi))/(m_A+m_B)= ((7\cdot 6)+(2 \cdot (- 12)))/(7+2)= (18)/(9)=2 m/s`
and the positive sign means the two balls are going to the right.


2) I assume here we are talking about an elastic collision. In this case, both total momentum and total kinetic energy are conserved:

`m_A v_(Ai)+m_B v_(Bi) = m_A v_(fA) + m_B v_(fB)`

`(1)/(2)m_A v_(Ai)^2+ (1)/(2)m_B v_(Bi)^2= (1)/(2)m_Av_(fA)^2+ (1)/(2)m_B v_(fB)^2`
where

`v_(fA)` is the final velocity of ball A

`v_(fB)` is the final velocity of ball B

If we solve simultaneously the two equations, we find:

`v_(fA)= (v_(Ai)(m_A-m_B)+2m_Bv_(Bi))/(m_A+m_B) = ((6)(7-2)+2(2)(-12))/(7+2)=-2 m/s`

`v_(fB)= (v_(Bi)(m_B-m_A)+2m_Av_(Ai))/(m_A+m_B) = ((-12)(2-7)+2(7)(6))/(7+2)= (144)/(9)=16 m/s`
So, after the collision, ball A moves to the left with velocity v=-2 m/s and ball B moves to the right with velocity v=16 m/s.

3) The total momentum before and after the collision is conserved.
In fact, the total momentum before the collision is:

`p_i = m_A v_(A) + m_B v_(fB) = (7\cdot 6)+(2 \cdot (-12))=42-24=18 m/s`
and the total momentum after the collision is:

`p_f = m_A v_(A) + m_B v_(fB) = (7\cdot (-2))+(2 \cdot 16)=-14+32=18 m/s`