What is the average force exerted by the plate on the ball

[mkwok]

Homework Statement

A 4.00 kg steel ball is dropped onto a copper plate from a height of 10.0 m. If the ball leaves a dent 2.75 mm deep, what is the average force exerted by the plate on the ball during the impact?

Homework Equations

KE=1/2mv^2
F=ma
Wgrav=mgh

The Attempt at a Solution

when the ball hit the plate, the ball is traveling at a speed of 2ax, which is also 196m/s
so the Kinetic Energy would be 76832 the moment the ball hits the plate

since W=Fd, so 76832=F(2.75/1000)
F=27938909.1N

just looking by the numbers, I can tell this is wrong

 

[rock.freak667]

Ignoring air resistance..all the gravitational potential energy is converted into kinetic energy as the mass falls...when the ball hits the copper plate, the work done in moving the ball 2.75mm into the copper. so that basically the loss in k.e. is equal to the work done in moving it into the copper
$W=Fs$ so that $mgh=Fs$

 

[ogb p]

.

Thank you for your question. In order to accurately calculate the average force exerted by the plate on the ball, we need to use the principle of conservation of energy. This means that the total energy of the system (ball and plate) before and after the impact should be equal.

Initially, the ball has only potential energy (due to its position) given by mgh, where m is the mass of the ball, g is the acceleration due to gravity and h is the initial height. In this case, m=4.00 kg, g=9.8 m/s^2 and h=10.0 m. Therefore, the initial potential energy is 392 J.

After the impact, the ball has both kinetic energy (due to its motion) and potential energy (due to its new position). The kinetic energy can be calculated using the equation KE=1/2mv^2, where m is the mass of the ball and v is the velocity just before impact. In this case, v=196 m/s (assuming no air resistance). Therefore, the kinetic energy is 76832 J.

The potential energy after the impact can be calculated using the same equation as before, but with a new height of 10.00275 m (initial height + dent depth). This gives a potential energy of 392.109 J.

Since the total energy before and after the impact should be equal, we can set up an equation:

Initial potential energy = Final kinetic energy + Final potential energy

392 J = 76832 J + 392.109 J

This gives a final kinetic energy of 76423.891 J.

Now, we can use the equation F=ma to calculate the average force exerted by the plate on the ball. We know the mass of the ball (4.00 kg) and we just calculated the final kinetic energy (76423.891 J). Therefore, the average force is:

F = ma = (4.00 kg)(76423.891 J) = 305695.564 N

So, the average force exerted by the plate on the ball during the impact is approximately 305695.564 N. I hope this helps clarify the concept for you.