A 0.240-kg billiard ball that is moving at 3.00 m/s strikes the bumper
of a pool table and bounces straight back at 2.40 m/s (80% of its original
speed). The collision lasts 0.0150 s. (a) Calculate the average force
exerted on the ball by the bumper. (b) How much kinetic energy in joules
is lost during the collision? (c) What percent of the original energy is left
23.4
To answer these questions, we need to apply the principles of conservation of momentum and conservation of kinetic energy.
(a) Calculate the average force exerted on the ball by the bumper:
The average force can be found using the impulse-momentum theorem, which states that the impulse experienced by an object is equal to the product of the force and the time interval over which it acts. The impulse can be calculated as the change in momentum of the ball.
The initial momentum of the ball is given by:
p_initial = m * v_initial
where m = mass of the ball and v_initial = initial velocity of the ball.
The final momentum of the ball is given by:
p_final = m * v_final
where v_final = final velocity of the ball.
The change in momentum is given by:
Δp = p_final - p_initial
The average force can be calculated by dividing the change in momentum by the time interval of the collision:
Average Force = Δp / Δt
Given:
m = 0.240 kg (mass of the ball)
v_initial = 3.00 m/s (initial velocity of the ball)
v_final = 2.40 m/s (final velocity of the ball)
Δt = 0.0150 s (time interval of the collision)
Now, let's calculate the average force:
p_initial = m * v_initial = 0.240 kg * 3.00 m/s = 0.720 kg·m/s
p_final = m * v_final = 0.240 kg * 2.40 m/s = 0.576 kg·m/s
Δp = p_final - p_initial = 0.576 kg·m/s - 0.720 kg·m/s = -0.144 kg·m/s
Average Force = Δp / Δt = (-0.144 kg·m/s) / (0.0150 s) = -9.6 N
Therefore, the average force exerted on the ball by the bumper is -9.6 N (negative sign indicates that the force is in the opposite direction of motion).
(b) Calculate the kinetic energy lost during the collision:
The initial kinetic energy of the ball is given by:
KE_initial = (1/2) * m * v_initial^2
The final kinetic energy of the ball is given by:
KE_final = (1/2) * m * v_final^2
The kinetic energy lost during the collision is given by:
KE_lost = KE_initial - KE_final
Let's calculate the kinetic energy lost:
KE_initial = (1/2) * m * v_initial^2 = (1/2) * 0.240 kg * (3.00 m/s)^2 = 0.540 J
KE_final = (1/2) * m * v_final^2 = (1/2) * 0.240 kg * (2.40 m/s)^2 = 0.3456 J
KE_lost = KE_initial - KE_final = 0.540 J - 0.3456 J = 0.1944 J
Therefore, the kinetic energy lost during the collision is 0.1944 J.
(c) Calculate the percentage of the original energy that is left:
The percentage of energy left can be calculated by dividing the final kinetic energy by the initial kinetic energy and multiplying by 100.
Percentage of Original Energy Left = (KE_final / KE_initial) * 100
Let's calculate the percentage of the original energy left:
Percentage of Original Energy Left = (0.3456 J / 0.540 J) * 100 = 64%
Therefore, 64% of the original energy is left after the collision.
(a) To calculate the average force exerted on the ball by the bumper, we can use the impulse-momentum theorem, which states that the change in momentum of an object is equal to the impulse applied to it. Mathematically, this is expressed as:
Impulse = Change in Momentum
The momentum of an object is given by the product of its mass and velocity:
Momentum = mass × velocity
The change in momentum is then:
Change in Momentum = Final Momentum - Initial Momentum
Given that the final velocity (vf) is -2.40 m/s (since it bounces straight back) and the initial velocity (vi) is 3.00 m/s, the change in velocity is:
Change in Velocity = vf - vi = -2.40 m/s - 3.00 m/s = -5.40 m/s
Substituting the mass (m = 0.240 kg) and the change in velocity into the equation, we have:
Impulse = Change in Momentum = mass × Change in Velocity
Impulse = 0.240 kg × (-5.40 m/s) = -1.296 kg·m/s (negative sign indicates the change in direction)
Since the impulse is equal to the average force multiplied by the time of collision, we can rearrange the equation to solve for the average force:
Average Force = Impulse / Time of Collision
Average Force = -1.296 kg·m/s / 0.0150 s = -86.4 N
Therefore, the average force exerted on the ball by the bumper is -86.4 N (negative sign indicates opposite direction).
(b) To find the amount of kinetic energy lost during the collision, we can calculate the initial kinetic energy (Ki) and the final kinetic energy (Kf), and then subtract the final kinetic energy from the initial kinetic energy:
Initial Kinetic Energy (Ki) = (1/2) × mass × (initial velocity)^2
Final Kinetic Energy (Kf) = (1/2) × mass × (final velocity)^2
Given that the mass (m) is 0.240 kg and the initial velocity (vi) is 3.00 m/s, we have:
Ki = (1/2) × 0.240 kg × (3.00 m/s)^2 = 0.540 J
Given that the final velocity (vf) is -2.40 m/s (since it bounces straight back), we have:
Kf = (1/2) × 0.240 kg × (-2.40 m/s)^2 = 0.345 J
Therefore, the kinetic energy lost during the collision is:
Kinetic Energy Lost = Ki - Kf = 0.540 J - 0.345 J = 0.195 J
(c) To find the percent of the original energy left, we can calculate the ratio of the final kinetic energy (Kf) to the initial kinetic energy (Ki), and then multiply by 100:
Percent Remaining = (Kf / Ki) × 100
Given that Kf is 0.345 J and Ki is 0.540 J, we have:
Percent Remaining = (0.345 J / 0.540 J) × 100 ≈ 63.9%
Therefore, approximately 63.9% of the original energy is left.