With her bare hands and her arms rigidly extended, a girl catches a 0.16kg baseball coming directly at her at speed of 25m/s. She shouts, “Ouch!’ because the ball stings her hands. She learns quickly how to move her hands with the ball as she catches it. If the contact time of the collision is increased from 3.5ms to 8.5ms in this manner, how do the magnitudes of the average impulse forces compare? Calculate these average forces.

Question

With her bare hands and her arms rigidly extended, a girl catches a 0.16kg baseball coming directly at her at speed of 25m/s. She shouts, “Ouch!’ because the ball stings her hands. She learns quickly how to move her hands with the ball as she catches it. If the contact time of the collision is increased from 3.5ms to 8.5ms in this manner, how do the magnitudes of the average impulse forces compare? Calculate these average forces.

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Answer 1

To calculate the average force, we can use the formula for impulse:

Impulse = Change in momentum = Force * Time

The initial momentum of the baseball is given by:

Initial momentum = mass * initial velocity = 0.16 kg * 25 m/s = 4 kg·m/s

The final momentum will be equal to the initial momentum since the girl catches the ball and brings it to a stop.

Final momentum = 4 kg·m/s

Using the formula for impulse, we can find the average force:

Initial impulse = Initial force * Initial time = Initial force * 3.5 ms

Final impulse = Final force * Final time = Final force * 8.5 ms

Since the momentum change is the same, we have:

Initial impulse = Final impulse

Initial force * 3.5 ms = Final force * 8.5 ms

Divide both sides by 3.5 ms:

Initial force = Final force * (8.5 ms / 3.5 ms)

Simplify:

Initial force = Final force * 2.43

Now we can solve for the average forces. Let's plug in the values:

Initial force * 3.5 ms = (Final force * 2.43) * 8.5 ms

Cancel out the time units:

Initial force * 3.5 = Final force * 2.43 * 8.5

Divide both sides by 3.5:

Initial force = Final force * 2.43 * 8.5 / 3.5

Calculate the average forces:

Initial force = Final force * 5.91

Now, we can find the average forces by choosing a value for the final force. For example, let's choose the final force to be 10 N:

Initial force = 10 N * 5.91 = 59.1 N

So, if the final force is 10 N, the initial force would be 59.1 N.

To compare the magnitudes of the average impulse forces, we can use the ratio of the forces:

Ratio = Initial force / Final force

For our example:

Ratio = 59.1 N / 10 N = 5.91

So, the magnitudes of the average impulse forces increased by a factor of 5.91 when the contact time of the collision increased from 3.5 ms to 8.5 ms.

Answer 2

The force=changemomentum/time

because the momentum change is the same, the force changes by 3.5/8.5 of the original.

average force=momentumchange/time
for the first one:
force=.16*25/.0035 N