A baseball (m = 135 g) approaches a bat horizontally at a speed of 40.4 m/s (90 mi/h) and is hit straight back at a speed of 45.7 m/s (102 mi/h). If the ball is in contact with the bat for a time of 1.00 ms, what is the average force exerted on the ball by the bat? Neglect the weight of the ball, since it is so much less than the force of the bat. Choose the direction of the incoming ball as the positive direction.
To find the average force exerted on the ball by the bat, we can use Newton's second law of motion, which states that force is equal to the change in momentum per unit of time.
First, let's find the initial momentum of the ball. The momentum of an object is given by the product of its mass and its velocity. In this case, we have:
Initial momentum (p_initial) = mass (m) * initial velocity (v_initial)
Where:
- mass (m) is given as 135 g, but we need to convert it to kilograms by dividing by 1000 (1 kg = 1000 g).
- The initial velocity (v_initial) is given as 40.4 m/s.
Therefore, the initial momentum (p_initial) = (0.135 kg) * (40.4 m/s).
Next, let's find the final momentum of the ball. We use the same formula, but with the final velocity (v_final) instead of initial velocity:
Final momentum (p_final) = mass (m) * final velocity (v_final)
Where:
- The mass (m) is the same as before.
- The final velocity (v_final) is given as 45.7 m/s.
Therefore, the final momentum (p_final) = (0.135 kg) * (45.7 m/s).
To find the change in momentum, we subtract the initial momentum from the final momentum:
Change in momentum (Δp) = p_final - p_initial.
Finally, we divide the change in momentum by the time of contact (t) to find the average force:
Average force (F) = Δp / t.
Where:
- The change in momentum (Δp) is obtained from the previous step.
- The time of contact (t) is given as 1.00 ms, but we need to convert it to seconds by dividing by 1000 (1 s = 1000 ms).
Solving these calculations will give you the average force exerted on the ball by the bat.