A superball with a mass of 71.2 g is dropped from a height of 2.5 m. It rebounds to a height of 2 m.
What is the change in its linear momen- tum during the collision with the floor? The acceleration of gravity is 9.8 m/s2 .
Answer in units of kg · m/s
To find the change in the linear momentum of the superball during the collision with the floor, we need to calculate the initial and final momentum and then subtract the initial momentum from the final momentum.
The momentum of an object is calculated by multiplying its mass by its velocity. Since we have the mass of the superball, we need to find its initial and final velocities.
Using the conservation of energy principle, we can find the initial velocity of the superball just before impact. The initial potential energy of the ball is converted into kinetic energy during the fall.
From the given height (2.5 m), we can calculate the potential energy:
Potential Energy = mass * gravity * height
= 0.0712 kg * 9.8 m/s^2 * 2.5 m
= 1.7456 J
The potential energy is converted into kinetic energy when the superball reaches the ground, so we can equate the two:
Potential Energy = Kinetic Energy
1.7456 J = (1/2) * mass * velocity^2
1.7456 J = (1/2) * 0.0712 kg * velocity^2
Now we can solve for the initial velocity (v_initial):
1.7456 J = (1/2) * 0.0712 kg * v_initial^2
v_initial^2 = (1.7456 J * 2) / (0.0712 kg)
v_initial^2 = 49.028 J/kg
v_initial = sqrt(49.028 J/kg)
v_initial ≈ 7 m/s
The final potential energy can be calculated in the same way using the final height (2 m):
Potential Energy = mass * gravity * height
= 0.0712 kg * 9.8 m/s^2 * 2 m
= 1.39232 J
Since the superball rebounds to this height, the kinetic energy will be equal to the potential energy before the collision:
Kinetic Energy = Potential Energy = 1.39232 J
Now we can solve for the final velocity (v_final):
1.39232 J = (1/2) * 0.0712 kg * v_final^2
v_final^2 = (1.39232 J * 2) / (0.0712 kg)
v_final^2 = 39.00672 J/kg
v_final = sqrt(39.00672 J/kg)
v_final ≈ 6.2 m/s
Finally, we can calculate the change in linear momentum:
Change in momentum = final momentum - initial momentum
= (mass * final velocity) - (mass * initial velocity)
= mass * (final velocity - initial velocity)
= 0.0712 kg * (6.2 m/s - 7 m/s)
= -0.05664 kg∙m/s
Therefore, the change in the linear momentum of the superball during the collision with the floor is approximately -0.05664 kg·m/s.