Ball 1, with a mass of 180 g and traveling at 18 m/s, collides head on with ball 2, which has a mass of 370 g and is initially at rest
(a) What are the final velocities of each ball if the collision is perfectly elastic?
(b) What are the final velocities of each ball if the collision is perfectly inelastic?
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
(a) In a perfectly elastic collision, both momentum and kinetic energy are conserved.
1. Let's start by calculating the initial and final momentum of each ball.
- Initial momentum of ball 1 = mass of ball 1 * velocity of ball 1
= 0.180 kg * 18 m/s = 3.24 kg⋅m/s
- Initial momentum of ball 2 = mass of ball 2 * velocity of ball 2
= 0.370 kg * 0 m/s = 0 kg⋅m/s
According to the conservation of momentum,
Total initial momentum = Total final momentum
Therefore, 3.24 kg⋅m/s + 0 kg⋅m/s = final momentum of ball 1 + final momentum of ball 2
Since ball 2 is initially at rest, the initial momentum is zero.
Therefore, we have:
3.24 kg⋅m/s = final momentum of ball 1
2. Now, we need to calculate the final velocities of each ball using the equation of momentum:
momentum = mass * velocity
Final velocity of ball 1 = final momentum of ball 1 / mass of ball 1
Final velocity of ball 1 = 3.24 kg⋅m/s / 0.180 kg = 18 m/s
Since ball 2 is initially at rest, it will take on the momentum of ball 1 in the opposite direction.
Final momentum of ball 2 = -final momentum of ball 1
Final velocity of ball 2 = final momentum of ball 2 / mass of ball 2
= -3.24 kg⋅m/s / 0.370 kg = -8.76 m/s
Therefore, the final velocities of the balls in a perfectly elastic collision are:
- Ball 1: 18 m/s
- Ball 2: -8.76 m/s
(b) In a perfectly inelastic collision, the two balls stick together after the collision, and momentum is conserved but kinetic energy is not.
1. Again, let's calculate the initial and final momentum of each ball.
- Initial momentum of ball 1 = 3.24 kg⋅m/s
- Initial momentum of ball 2 = 0 kg⋅m/s
According to the conservation of momentum,
Total initial momentum = Total final momentum
Therefore, 3.24 kg⋅m/s + 0 kg⋅m/s = final momentum of the combined balls
2. To calculate the final velocity of the combined balls, we need to know their combined mass.
Combined mass = mass of ball 1 + mass of ball 2
= 0.180 kg + 0.370 kg
= 0.550 kg
Since the balls stick together, the final momentum of the combined balls is the same as their initial momentum.
Final velocity of the combined balls = final momentum of the combined balls / combined mass
= 3.24 kg⋅m/s / 0.550 kg
≈ 5.89 m/s
Therefore, in a perfectly inelastic collision, the final velocity of the combined balls is approximately 5.89 m/s.
To find the final velocities of each ball in these scenarios, we can use the principles of conservation of momentum and conservation of kinetic energy.
(a) In a perfectly elastic collision, both momentum and kinetic energy are conserved.
1. Start by finding the initial momentum of each ball:
- Momentum (p) is given by the formula p = mass × velocity.
- For ball 1: p₁ = m₁v₁ = (0.180 kg) × (18 m/s) = 3.24 kg·m/s.
- Since ball 2 is initially at rest, its initial momentum is zero: p₂ = 0 kg·m/s.
2. Apply the conservation of momentum:
- In a perfectly elastic collision, the total initial momentum is equal to the total final momentum.
- Therefore, p₁ + p₂ = p₁' + p₂' (where ' denotes the final velocities).
- Substituting the values, we have 3.24 kg·m/s + 0 kg·m/s = p₁' + p₂'.
3. Since ball 2 is at rest initially (zero velocity), its final velocity will depend on the mass ratio between ball 1 and ball 2.
- The mass ratio is given by the formula m₁/m₂.
- For our scenario, m₁ = 180 g (0.180 kg) and m₂ = 370 g (0.370 kg), so the mass ratio is roughly 0.486.
4. Using the mass ratio, we can calculate the final velocities of the balls:
- Rearranging the momentum equation, we get p₁' = 3.24 kg·m/s - (0.486) × p₂'.
- Since p₂' = 0 kg·m/s (initially at rest), we can simplify the equation to p₁' = 3.24 kg·m/s.
- Dividing p₁' by the mass of ball 1, we get the final velocity of ball 1: v₁' = p₁' / m₁.
- v₁' = (3.24 kg·m/s) / (0.180 kg) = 18 m/s.
- For ball 2, since its initial velocity is zero, the final velocity remains zero: v₂' = 0 m/s.
Therefore, in a perfectly elastic collision, ball 1 will rebound with the same initial velocity (18 m/s) and ball 2 will remain at rest (0 m/s).
(b) In a perfectly inelastic collision, only the momentum is conserved, while the kinetic energy is not.
1. Again, start by finding the initial momentum of each ball (same as in part a):
- For ball 1: p₁ = m₁v₁ = (0.180 kg) × (18 m/s) = 3.24 kg·m/s.
- For ball 2, initially at rest: p₂ = 0 kg·m/s.
2. Apply the conservation of momentum:
- In a perfectly inelastic collision, the total initial momentum is equal to the total final momentum.
- Therefore, p₁ + p₂ = (p₁' + p₂') (where ' denotes the final velocities).
- Substituting the values, we have 3.24 kg·m/s + 0 kg·m/s = p₁' + p₂'.
3. Since the two balls stick together after the collision, they travel with the same final velocity v' as a combined mass.
- The combined mass can be calculated by summing the individual masses: M = m₁ + m₂.
- For our scenario, M = 180 g (0.180 kg) + 370 g (0.370 kg) = 0.550 kg.
4. Using the combined mass, we can calculate the final velocities of the balls:
- Since they stick together, their final velocities are the same as the combined velocity v': p₁' + p₂' = M × v'.
- Now, substitute p₁' + p₂' = 3.24 kg·m/s:
- 3.24 kg·m/s = (0.550 kg) × v'.
- Solving for v', we get v' = 3.24 kg·m/s / (0.550 kg) = 5.90 m/s.
Therefore, in a perfectly inelastic collision, both balls will stick together and move with a final velocity of 5.90 m/s.