A steel ball is dropped onto a hard floor from a height of 1.80 m and rebounds to a height of 1.64 m. (Assume that the positive direction is upward.)
(a) Calculate its velocity just before it strikes the floor.
m/s
(b) Calculate its velocity just after it leaves the floor on its way back up.
m/s
(c) Calculate its acceleration during contact with the floor if that contact lasts 0.0800 ms.
m/s2
(d) How much did the ball compress during its collision with the floor, assuming the floor is absolutely rigid?
m
To solve this problem, we can use the concepts of conservation of energy and impulse-momentum.
(a) To calculate the velocity just before the ball strikes the floor, we can use the principle of conservation of mechanical energy. The initial potential energy is converted into kinetic energy just before it hits the floor.
First, we need to determine the potential energy at the initial height of 1.8 m:
Potential Energy = mass * gravity * height
We need the mass of the ball and the acceleration due to gravity. Let's assume the mass of the ball is 1 kg and acceleration due to gravity is 9.8 m/s^2.
Potential Energy = 1 kg * 9.8 m/s^2 * 1.8 m
Next, we equate the potential energy to the kinetic energy just before it strikes the floor:
Potential Energy = Kinetic Energy
1/2 * mass * velocity^2 = 1 kg * 9.8 m/s^2 * 1.8 m
Now we can solve for the velocity:
velocity^2 = 2 * 9.8 m/s^2 * 1.8 m
velocity = √(2 * 9.8 m/s^2 * 1.8 m)
Substituting the values and calculating the square root, we get the velocity just before it strikes the floor.
(b) To calculate the velocity just after it leaves the floor on its way back up, we can use the principle of conservation of mechanical energy again. This time, we consider the potential energy at the height of 1.64 m.
Use the same formula as in part (a) to find the potential energy at 1.64 m and then equate it to the kinetic energy just after it leaves the floor.
(c) To calculate the acceleration during contact with the floor, we can use the concept of impulse-momentum. Impulse is defined as the change in momentum, and it is equal to the force applied multiplied by the contact time.
Impulse = force * contact time
Change in momentum = mass * change in velocity
Acceleration = Change in velocity / contact time
First, calculate the change in velocity by subtracting the velocity just before and just after the collision:
Change in velocity = velocity just after - velocity just before
Next, calculate the acceleration using the formula:
Acceleration = Change in velocity / contact time
(d) To find the compression of the ball during its collision with the floor, we need to know the coefficient of restitution, which represents the elasticity of the collision. Assuming the floor is absolutely rigid means the collision is perfectly inelastic, with a coefficient of restitution of zero.
The compression can be calculated using the formula:
Compression = initial height - final height
Substituting the given values, we can calculate the compression.