A ball of mass 0.150kg is dropped from a height of 1.25m. It rebounds from the floor to reach a height of 0.960m. What impulse was given to the ball by the floor?
Calculate the downward velocity V1 when it hits the floor:
V1 = sqrt (2gH), where H = 1.25 m
Then calculate the upward velocity after the bounce:
V2 = sqrt (2gH'), where H' = 0.96 m
The momentum change is M(V1 + V2), since the V's are in opposite directions.
The momentum change equals the impulse given by the floor.
1.35kgm/s
To find the impulse given to the ball by the floor, we can use the principle of conservation of energy. Impulse can be calculated by using the difference in the kinetic energies of the ball before and after impact.
First, let's calculate the initial potential energy of the ball before it is dropped:
Potential Energy (PE) = mass * gravity * height
PE = 0.150kg * 9.8m/s^2 * 1.25m
PE = 1.8385 J (Joules)
Next, let's calculate the final potential energy of the ball at a height of 0.960m:
PE = 0.150kg * 9.8m/s^2 * 0.960m
PE = 1.4112 J
Since there is no loss or gain of energy in an idealized system, the change in potential energy is equal to the change in kinetic energy:
Change in kinetic energy = Initial kinetic energy - Final kinetic energy
Since the ball is at rest before it is dropped, the initial kinetic energy is 0.
Therefore, Change in kinetic energy = Final kinetic energy - 0
Final kinetic energy = PE - 0
Final kinetic energy = 1.4112 J
Now, we know that impulse (J) is equal to the change in kinetic energy:
J = Final kinetic energy
J = 1.4112 J
Therefore, the impulse given to the ball by the floor is equal to 1.4112 Joules.
To calculate the impulse given to the ball by the floor, we need to use the principle of conservation of mechanical energy. The mechanical energy of the system (ball) is conserved before and after the collision with the floor.
The mechanical energy of the ball consists of its kinetic energy (KE) and potential energy (PE). At the initial height, all its energy is potential energy, and at the highest point of rebound, all its energy is again potential energy. This is because the ball momentarily comes to rest at its highest point, meaning its kinetic energy is zero.
The total mechanical energy of the ball before the collision is given by its potential energy at the initial height:
PE_initial = m * g * h_initial
Where:
m = mass of the ball (0.150 kg)
g = acceleration due to gravity (9.8 m/s^2)
h_initial = initial height (1.25 m)
The total mechanical energy of the ball after the collision is given by its potential energy at the height of rebound:
PE_final = m * g * h_final
Where:
h_final = height of rebound (0.960 m)
Since mechanical energy is conserved, we can equate the initial and final potential energies:
PE_initial = PE_final
Therefore:
m * g * h_initial = m * g * h_final
Now, we need to solve for the impulse given to the ball by the floor. Impulse is defined as the change in momentum of an object. In this case, the momentum change is caused by the floor's exertion of a force on the ball over a specific time interval.
Using the impulse-momentum principle, the impulse (J) can be calculated as:
J = m * Δv
Where:
Δv = change in velocity of the ball
Since the ball rebounds, its change in velocity can be written as:
Δv = v_final - v_initial
The initial velocity (v_initial) is the velocity just before the collision with the floor, and it can be calculated using the equation for the conservation of mechanical energy:
v_initial = √(2 * g * h_initial)
The final velocity (v_final) is the velocity just after the collision with the floor, which can be calculated as:
v_final = -√(2 * g * h_final)
The negative sign indicates a change in direction after the rebound.
Let's now calculate the impulse given to the ball by the floor.
1. Calculate the initial velocity:
v_initial = √(2 * 9.8 * 1.25)
2. Calculate the final velocity:
v_final = -√(2 * 9.8 * 0.960)
3. Calculate the change in velocity:
Δv = v_final - v_initial
4. Calculate the impulse:
J = m * Δv
Substituting the given values:
J = (0.150 kg) * (v_final - v_initial)
Evaluate the expression to find the impulse given to the ball by the floor.