A baseball (m = 149 g) approaches a bat horizontally at a speed of 42.2 m/s (94 mi/h) and is hit straight back at a speed of 48.1 m/s (108 mi/h). If the ball is in contact with the bat for a time of 1.10 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 the impulse-momentum principle. The impulse experienced by the ball is equal to the change in its momentum.
1. Calculate the initial momentum of the ball before impact:
momentum = mass × velocity_1
momentum = 149 g × 42.2 m/s
2. Calculate the final momentum of the ball after impact:
momentum = mass × velocity_2
momentum = 149 g × (-48.1 m/s)
(Note: Since the ball is hit straight back, the final velocity is in the opposite direction, hence the negative sign.)
3. Calculate the change in momentum:
change in momentum = final momentum - initial momentum
4. Calculate the impulse experienced by the ball:
impulse = change in momentum
5. Finally, calculate the average force exerted on the ball by dividing the impulse by the contact time:
average force = impulse / contact time
Let's plug in the values and calculate:
1. Initial momentum:
momentum = 149 g × 42.2 m/s = 0.149 kg × 42.2 m/s
2. Final momentum:
momentum = 149 g × (-48.1 m/s) = 0.149 kg × (-48.1 m/s)
3. Change in momentum:
change in momentum = final momentum - initial momentum
4. Impulse:
impulse = change in momentum
5. Average force:
average force = impulse / contact time
Plug in the calculated values and solve for the average force exerted on the ball.
To find the average force exerted on the ball by the bat, we can use the impulse-momentum principle, which states that the change in momentum of an object is equal to the force applied to it multiplied by the time for which the force is applied.
First, let's determine the initial and final momenta of the ball.
The initial momentum (p_initial) of the ball can be calculated by multiplying its mass (m) by its initial velocity (v_initial):
p_initial = m * v_initial
Similarly, the final momentum (p_final) of the ball can be calculated by multiplying its mass (m) by its final velocity (v_final):
p_final = m * v_final
Now, let's find the change in momentum (Δp) using the formula:
Δp = p_final - p_initial
Next, let's calculate the change in momentum in terms of velocity units. We can convert the velocities from m/s to mi/h to ensure consistent units:
Δp = (p_final - p_initial) * (1 mi/h) / (0.44704 m/s)
Next, let's convert the time of contact from milliseconds (ms) to seconds (s):
t = 1.10 ms * (1 s / 1000 ms)
Finally, we can calculate the average force (F_avg) exerted on the ball using the impulse-momentum principle:
F_avg = Δp / t
Let's plug in the values and calculate the average force:
m = 149 g = 149/1000 kg
v_initial = 42.2 m/s
v_final = -48.1 m/s (negative because the ball is hit back in the opposite direction)
t = 1.10 ms * (1 s / 1000 ms)
p_initial = m * v_initial
p_final = m * v_final
Δp = (p_final - p_initial) * (1 mi/h) / (0.44704 m/s)
F_avg = Δp / t