A racquet ball with mass m = 0.238 kg is moving toward the wall at v = 12.5 m/s and at an angle of θ = 31° with respect to the horizontal. The ball makes a perfectly elastic collision with the solid, frictionless wall and rebounds at the same angle with respect to the horizontal. The ball is in contact with the wall for t = 0.078 s.
a. What is the magnitude of the initial momentum of the racquet ball?
b. What is the magnitude of the change in momentum of the racquet ball?
c. What is the magnitude of the average force the wall exerts on the racquet ball?
d. Now the racquet ball is moving straight toward the wall at a velocity of vi = 12.5 m/s. The ball makes an inelastic collision with the solid wall and leaves the wall in the opposite direction at vf = -7.9 m/s. The ball exerts the same average force on the ball as before.
What is the magnitude of the change in momentum of the racquet ball?
e. What is the time the ball is in contact with the wall?
f. What is the change in kinetic energy of the racquet ball?
(a) magnitude of the initial momentum is
(0.238 kg)(12.5 m/s) = 2.975 kg*m/s.
(b) The horizontal component of the initial momentum is
(0.238 kg)(12.5 m/s)*cos(31) = 2.550 kg*m/s.
The change in momentum is -2*2.550 kg*m/s, so the magnitude of the change is 5.10 kg*m/s.
(c) average F = delta-p/delta-t
= (5.10 kg*m/s)/(0.078 s) = 65.4 N.
(d) |delta-p| = (20.4 m/s)*(0.238 kg)
= 4.855 kg*m/s.
(e) t = (4.855 kg*m/s)/(65.4 N) = 0.0742 s.
(f) Change in KE is
(1/2)(0.238 kg)[(7.9 m/s)^2 - (12.5 m/s)^2]
= -11.17 J
a. The magnitude of the initial momentum of the racquet ball can be calculated using the formula:
P = m * v
where m is the mass of the ball and v is its velocity. Substituting the given values, we have:
P = 0.238 kg * 12.5 m/s ≈ 2.975 kg*m/s
b. The magnitude of the change in momentum can be calculated using the formula:
ΔP = 2P
where P is the initial momentum. Substituting the value from part a, we have:
ΔP = 2 * 2.975 kg*m/s ≈ 5.95 kg*m/s
c. The magnitude of the average force the wall exerts on the racquet ball can be calculated using the formula:
F = ΔP / t
where ΔP is the change in momentum and t is the time the ball is in contact with the wall. Substituting the given values, we have:
F = 5.95 kg*m/s / 0.078 s ≈ 76.28 N
d. In an inelastic collision, the magnitude of the change in momentum can be calculated using the formula:
ΔP = m * |vf - vi|
where m is the mass of the ball, vf is the final velocity, and vi is the initial velocity. Substituting the given values, we have:
ΔP = 0.238 kg * |-7.9 m/s - 12.5 m/s| ≈ 5.868 kg*m/s
e. The time the ball is in contact with the wall can be calculated using the formula:
t = ΔP / F
where ΔP is the change in momentum and F is the average force. Substituting the given values, we have:
t = 5.868 kg*m/s / 76.28 N ≈ 0.077 s
f. The change in kinetic energy of the racquet ball can be calculated using the formula:
ΔKE = 0.5 * m * (vf^2 - vi^2)
where m is the mass of the ball, vf is the final velocity, and vi is the initial velocity. Substituting the given values, we have:
ΔKE = 0.5 * 0.238 kg * ((-7.9 m/s)^2 - (12.5 m/s)^2) ≈ -38.106 J
To solve these questions step-by-step, we'll need to use the principles of momentum and kinetic energy.
a. The magnitude of the initial momentum of the racquet ball can be calculated using the formula p = m * v, where m is the mass and v is the velocity of the ball. In this case, the mass m = 0.238 kg and the velocity v = 12.5 m/s.
Initial momentum, p = 0.238 kg * 12.5 m/s = 2.975 kg·m/s
b. The magnitude of the change in momentum can be calculated using the principle of conservation of momentum. Since the ball rebounds at the same angle with the same magnitude of velocity, the change in momentum is twice the magnitude of the initial momentum.
Change in momentum = 2 * 2.975 kg·m/s = 5.95 kg·m/s
c. The magnitude of the average force the wall exerts on the racquet ball can be calculated using the impulse-momentum theorem. The impulse is given by J = F * t, where J is the change in momentum, F is the average force, and t is the time of contact.
Average force, F = J / t = (5.95 kg·m/s) / (0.078 s) = 76.28 N
d. In an inelastic collision, the two bodies stick together after the collision. The magnitude of the change in momentum is equal to the magnitude of the initial momentum, but in the opposite direction.
Magnitude of the change in momentum = 2 * 2.975 kg·m/s = 5.95 kg·m/s
e. The time the ball is in contact with the wall can be calculated using the formula t = 2 * v / a, where v is the initial velocity and a is the acceleration of the ball during the collision.
Since it is an inelastic collision, the acceleration can be calculated using the formula a = (vf - vi) / t, where vf is the final velocity and vi is the initial velocity.
Acceleration, a = (-7.9 m/s - 12.5 m/s) / (0.078 s) = -259.74 m/s^2
Time, t = 2 * 12.5 m/s / 259.74 m/s^2 = 0.096 s
f. The change in kinetic energy of the racquet ball can be calculated using the formula ΔKE = KEf - KEi, where KEf is the final kinetic energy and KEi is the initial kinetic energy.
Initial kinetic energy, KEi = 1/2 * m * v^2 = 1/2 * 0.238 kg * (12.5 m/s)^2 = 29.5625 J
Final kinetic energy, KEf = 1/2 * m * vf^2 = 1/2 * 0.238 kg * (-7.9 m/s)^2 = 7.0361 J
Change in kinetic energy, ΔKE = KEf - KEi = 7.0361 J - 29.5625 J = -22.5264 J
Note: The negative sign indicates a decrease in kinetic energy.
To solve these problems, we can use the principles of momentum and Newton's laws of motion.
a. The magnitude of the initial momentum of the racquet ball can be calculated using the formula:
Initial momentum = mass * velocity
Initial momentum = 0.238 kg * 12.5 m/s
b. The magnitude of the change in momentum can be calculated using the formula:
Change in momentum = Final momentum - Initial momentum
In this case, since the ball rebounds with the same angle and velocity, the final momentum is equal to the negative of the initial momentum. Therefore,
Change in momentum = -(2 * Initial momentum)
c. The magnitude of the average force the wall exerts on the racquet ball can be calculated using the formula:
Average force = Change in momentum / Time
In this case, the change in momentum has already been calculated in part b, and the time is given as 0.078 s.
d. Since the ball makes an inelastic collision with the wall and leaves in the opposite direction, we know that the final momentum is equal to the negative initial momentum, just as in part b. Therefore, the magnitude of the change in momentum is the same as in part b.
e. The time the ball is in contact with the wall, also known as the contact time, can be calculated using the formula:
Contact time = Change in momentum / (Average force * mass)
In this case, the change in momentum has already been calculated in part b, and the average force is given.
f. The change in kinetic energy of the racquet ball can be calculated using the formula:
Change in kinetic energy = Final kinetic energy - Initial kinetic energy
The initial kinetic energy can be calculated using the formula:
Initial kinetic energy = 0.5 * mass * (initial velocity)^2
The final kinetic energy can be calculated using the formula:
Final kinetic energy = 0.5 * mass * (final velocity)^2
The change in kinetic energy is then calculated by subtracting the initial kinetic energy from the final kinetic energy.