Instead of standing in the back of the lane and rolling the ball toward the pins like most people, you place a 5kg bowling ball at rest at the back of the lane and try to hit it toward the pins with a perfectly elastic collision using a 1kg ball rolling at 5 m/s from a 25 degree angle from the horizontal. Your perfect spin sends the larger ball flying or at least moving straight down the center of the lane toward the pin. What will the velocity of the larger bowling ball be?

Question

Instead of standing in the back of the lane and rolling the ball toward the pins like most people, you place a 5kg bowling ball at rest at the back of the lane and try to hit it toward the pins with a perfectly elastic collision using a 1kg ball rolling at 5 m/s from a 25 degree angle from the horizontal. Your perfect spin sends the larger ball flying or at least moving straight down the center of the lane toward the pin. What will the velocity of the larger bowling ball be?

Answer 1

To determine the velocity of the larger bowling ball after the collision, we can use the principle of conservation of momentum and the laws of physics.

1. Calculate the initial momentum of the smaller ball:
The initial momentum (p1) of the smaller ball can be calculated using the formula p = m * v, where:
- m is the mass of the smaller ball (1 kg)
- v is the velocity of the smaller ball (5 m/s)

p1 = (1 kg) * (5 m/s) = 5 kg * m/s

2. Determine the final momentum of the smaller ball:
Since the collision is perfectly elastic, meaning kinetic energy is conserved, the final momentum (p1') of the smaller ball will also be 5 kg * m/s.

3. Calculate the initial momentum of the larger ball:
The initial momentum (p2) of the larger ball is zero since it is initially at rest.

p2 = 0 kg * m/s

4. Determine the final momentum of the larger ball:
According to the conservation of momentum, the total momentum before the collision should equal the total momentum after the collision.

p1 + p2 = p1' + p2'

5 kg * m/s + 0 kg * m/s = 5 kg * m/s + p2'

p2' = 0 kg * m/s

Therefore, the velocity of the larger bowling ball after the collision will be zero (0 m/s).

Answer 2

To calculate the velocity of the larger bowling ball after the collision, we will use the principles of conservation of momentum and conservation of kinetic energy.

1. Calculate the initial momentum of the smaller ball:
The initial momentum (P₁) of the smaller ball can be found using the equation:
P₁ = mass₁ * velocity₁
P₁ = 1 kg * 5 m/s

2. Calculate the initial momentum of the larger ball:
Since the larger ball is initially at rest (not moving), its initial momentum (P₂) is zero.

3. Calculate the total initial momentum:
The total initial momentum (P_total) is given by the sum of the individual momenta:
P_total = P₁ + P₂

4. Solve for the velocity of the larger ball:
After the collision, the smaller ball will transfer its momentum to the larger ball in a perfectly elastic collision. This means that the total momentum before and after the collision remains constant.
Since the initial momentum of the larger ball is zero and the total momentum is conserved, we can equate the initial and final momenta:
P_total = P₃
P₃ = P₁ + P₂
0 = 1 kg * 5 m/s + 5 kg * v₃

Solving for v₃ (the velocity of the larger ball):
5 kg * v₃ = -1 kg * 5 m/s
v₃ = -1 m/s

Therefore, the velocity of the larger bowling ball after the collision would be -1 m/s, indicating that the larger ball is moving in the opposite direction with a speed of 1 m/s.