A billiard ball rolling across a table to the right at 2.2 m/s makes a head-on elastic collision with an identical ball. The mass of a billiard ball is 36 g.
If the second ball is initially at rest, what is the velocity of the first ball after the collision?
If the second ball is initially at rest, what is the velocity of the second ball after the collision?
If the second ball is initially moving to the left with a velocity of -1.1 m/s, what is the velocity of the first ball after the collision?
If the second ball is initially moving to the left with a velocity of -1.1 m/s, what is the velocity of the second ball after the collision?
To solve these questions, we can use the principles of conservation of momentum and conservation of kinetic energy.
First, let's calculate the initial momentum of the first ball (mass = 0.036 kg) when it's moving to the right at 2.2 m/s:
Momentum (P) = Mass (m) * Velocity (v)
P1 = (0.036 kg) * (2.2 m/s)
P1 = 0.0792 kg·m/s
Now, let's solve each question step by step:
1. If the second ball is initially at rest, what is the velocity of the first ball after the collision?
Since the second ball is initially at rest, the initial momentum of the system is just P1. During the elastic collision, momentum is conserved, so:
P1 = P1' + P2'
0.0792 kg·m/s = P1' + 0
The second ball remains at rest ("0" because it has no momentum initially), so the velocity of the first ball after the collision is 2.2 m/s to the right.
2. If the second ball is initially at rest, what is the velocity of the second ball after the collision?
Since the second ball is initially at rest, it acquires the entire momentum of the first ball during the collision. Therefore, the velocity of the second ball after the collision is also 2.2 m/s to the right.
3. If the second ball is initially moving to the left with a velocity of -1.1 m/s, what is the velocity of the first ball after the collision?
Now, let's calculate the initial momentum of the system:
P_initial = P1 + P2
P_initial = (0.036 kg) * (2.2 m/s) + (0.036 kg) * (-1.1 m/s)
P_initial = 0.0792 kg·m/s - 0.0396 kg·m/s
P_initial = 0.0396 kg·m/s
During the elastic collision, momentum is conserved:
P_initial = P1' + P2'
0.0396 kg·m/s = P1' + P2'
Since the second ball is initially moving to the left, its initial momentum is:
P2 = Mass (m) * Velocity (v)
P2 = (0.036 kg) * (-1.1 m/s)
P2 = -0.0396 kg·m/s
So, we can rewrite the equation:
0.0396 kg·m/s = P1' - 0.0396 kg·m/s
Solving for P1', the velocity of the first ball after the collision:
P1' = 0.0396 kg·m/s + 0.0396 kg·m/s
P1' = 0.0792 kg·m/s
Since the mass of the first ball remains the same, its velocity after the collision is 2.2 m/s to the right, just like in the first question.
4. If the second ball is initially moving to the left with a velocity of -1.1 m/s, what is the velocity of the second ball after the collision?
Since the second ball is initially moving with a velocity of -1.1 m/s to the left, its momentum is -0.0396 kg·m/s. During the elastic collision, momentum is conserved, so:
P_initial = P1' + P2'
0.0396 kg·m/s = P1' - 0.0396 kg·m/s
Solving for P1':
P1' = 0.0396 kg·m/s + 0.0396 kg·m/s
P1' = 0.0792 kg·m/s
Now, let's calculate the velocity of the second ball after the collision:
P2' = P_initial - P1'
P2' = 0.0396 kg·m/s - 0.0792 kg·m/s
P2' = -0.0396 kg·m/s
The negative sign indicates that the second ball is moving to the left with a velocity of -1.1 m/s, just like its initial velocity.
To summarize the answers:
1. The velocity of the first ball after the collision is 2.2 m/s to the right.
2. The velocity of the second ball after the collision is 2.2 m/s to the right.
3. The velocity of the first ball after the collision is 2.2 m/s to the right.
4. The velocity of the second ball after the collision is -1.1 m/s to the left.