Final answer:
The basketball's contact time with the backboard can be found by calculating the change in its momentum and using the impulse-momentum theorem. The calculation results in a contact time of approximately 0.0339 seconds.
Step-by-step explanation:
To find out how long the basketball was in contact with the backboard, we first need to calculate the change in momentum, which is the product of the mass and the change in velocity. Since the basketball bounces off the backboard, its velocity changes direction, and we need to take this into account when calculating. We begin by determining the initial and final momenta: the initial momentum is (0.48 kg) × (27.5 m/s), and the final momentum is (0.48 kg) × (-21.0 m/s), with the negative sign indicating the change in direction.
Now, the change in momentum (Δp) is the final momentum minus the initial momentum. Next, we use the impulse-momentum theorem which states that the change in momentum is equal to the impulse applied to it. Impulse is the product of force (F) and time (Δt). Given the force as 686 N, we solve for the time Δt.
Here are the steps in mathematical form:
- Initial momentum (pi): (0.48 kg) × (27.5 m/s) = 13.2 kg×m/s
- Final momentum (pf): (0.48 kg) × (-21.0 m/s) = -10.08 kg×m/s
- Change in momentum (Δp): -10.08 kg×m/s - 13.2 kg×m/s = -23.28 kg×m/s
- Impulse (J): 686 N × Δt
- Set the impulse equal to the change in momentum: 686 N × Δt = -23.28 kg×m/s
- Solve for Δt: Δt = -23.28 kg×m/s / 686 N = 0.0339 s
Therefore, the ball was in contact with the backboard for approximately 0.0339 seconds.