After falling from rest from a height of 34 m, a 0.47 kg ball rebounds upward, reaching a height of 24 m. If the contact between ball and ground lasted 1.8 ms, what average force was exerted on the ball?

the velocity of the ball at the ground was

vf^2=2gh

vf=sqrt 2gh.

but this velocity reversed at impact, so the change in velocity is
2sqrt2gh

force*time=2sqrt(2gh)

To find the average force exerted on the ball, we can use the principle of conservation of energy.

First, let's calculate the gravitational potential energy of the ball when it is at a height of 34 m (initial position) and when it reaches a height of 24 m (final position).

Gravitational potential energy (PE) is given by the formula:
PE = m * g * h
where:
m = mass of the ball (0.47 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height

Initial potential energy (PE) = 0.47 kg * 9.8 m/s^2 * 34 m
Final potential energy (PE) = 0.47 kg * 9.8 m/s^2 * 24 m

Next, we need to calculate the work done on the ball during the collision with the ground. Since the force acting on the ball is constant during the collision, we can use the work-energy theorem.

The work done (W) is given by the formula:
W = F * d * cos(theta)
where:
F = force exerted on the ball during the collision (unknown)
d = displacement during the collision (unknown)
theta = angle between the force and the displacement (assume 0 degrees, as the force is directed upward)

Since the ball rebounds upward and reaches a height of 24 m, the displacement during the collision is twice the height, 2 * 24 m.

Now, we can equate the initial potential energy to the sum of the final potential energy and the work done during the collision:

Initial PE = Final PE + Work done during collision
0.47 kg * 9.8 m/s^2 * 34 m = 0.47 kg * 9.8 m/s^2 * 24 m + F * 2 * 24 m

Simplifying the equation will give us the value of the unknown force (F).

Now, let's calculate the average force:
Average force = Total impulse / Time of contact

Impulse is the change in momentum. In this case, the ball comes to rest in the upward direction and then reverses its direction. So, the change in momentum is two times the momentum of the ball just before it hits the ground.

Momentum is given by the formula:
Momentum = mass * velocity

We know the mass of the ball (0.47 kg), and the velocity just before hitting the ground can be calculated using the time of contact (1.8 ms) and the displacement (2 * 24 m).

Finally, we can calculate the average force by dividing the total impulse by the time of contact.