After falling from rest from a height of 30 m, a 0.49 kg ball rebounds upward, reaching a height of 20 m. If the contact between ball and ground lasted 2.5 ms, what average force was exerted on the ball?
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To find the average force exerted on the ball during its contact with the ground, we can use the concept of impulse.
Impulse is the change in momentum of an object and is defined as the product of force and the time interval over which the force acts. Mathematically, it can be expressed as:
Impulse = Force × Time
Using this equation, we can find the average force exerted on the ball during the contact by calculating the impulse.
Step 1: Find the momentum before and after the collision.
Momentum is defined as the product of mass and velocity.
The initial momentum of the ball before the collision = mass × initial velocity
The final momentum of the ball after the collision = mass × final velocity
Given:
Mass of the ball (m) = 0.49 kg
Initial velocity (u) = 0 m/s (since the ball was at rest)
Final velocity (v) = ? (to be determined)
Using the principle of conservation of energy, we can determine the final velocity by considering the change in gravitational potential energy:
Initial potential energy = Final potential energy
mgh1 = mgh2
0.49 kg × 9.8 m/s^2 × 30 m = 0.49 kg × 9.8 m/s^2 × 20 m
This equation allows us to solve for h2, which is the height at which the ball rebounds (20 m). Now, using the equation for potential energy, we can determine the final velocity:
v = √(2gh2)
v = √(2 × 9.8 m/s^2 × 20 m)
v ≈ 19.80 m/s
The initial momentum (before collision) = 0.49 kg × 0 m/s = 0 kg·m/s
The final momentum (after collision) = 0.49 kg × 19.80 m/s
Step 2: Calculate the change in momentum (impulse).
Change in momentum = Final momentum - Initial momentum
Change in momentum = (0.49 kg × 19.80 m/s) - 0 kg·m/s = 9.702 kg·m/s
Step 3: Calculate the average force exerted on the ball.
Average force = Impulse / Time
Average force = (9.702 kg·m/s) / (2.5 × 10^-3 s)
Using a calculator, we can find that the average force exerted on the ball is approximately:
Average force ≈ 3,881.76 N
Therefore, the average force exerted on the ball during its contact with the ground is approximately 3,881.76 Newtons.