Sledge Hammer Demo?

Question

In the classroom demonstration where a volunteer strikes an anvil resting on the physics professor's chest will he lies reclined, the anvil shields the daring physics professor from most of the sledgehammer's:

a) momentum
b) kinetic energy
c)...both
d)...actually neither

The answer is b. Every bit of momentum imparted to the anvil by the sledge is imparted to the professor (and subsequently to the earth that supports him). The anvil doesn't shield the professor from the sledge's momentum -- not a bit. The shielding of kinetic energy is a different story. A significant fraction of the sledge's kinetic energy never gets to the professor -- it is absorbed by the anvil in the form of heat. Have you ever noticed that a hammer head gets warm after you have been hammering vigorously? Heat is the graveyard of kinetic energy.

We can be a bit more insightful about this and investigate the goings on during the impact between the sledge and anvil. During impact the force on the anvil at every instant is equal and opposite to the force on the sledge at the same instant. The sledge acts on the anvil just as long and just as hard as the anvil acts (reacts) on the sledge. Therefore the impulse or punch that stops the sledge is exactly equal to the impulse or...

punch that goes into the anvil, and then into the professor. if the sledge comes to a dead stop, the impulse must have cancelled all the sledge's momentum, and that same impulse must put the same amount of momentum into the anvil. So we see that the anvil gets every bit of the momentum the sledge loses -- the momentum is completely transferred from the sledge to the anvil. Of course, the newly acquired momentum does not make the anvil move very fast because it has much more mass than the sledge.

Now consider kinetic energy. When we analyze momentum we think about the time during which forces act, but when we analyze energy we think about the distance through which forces act. That's because the energy a body acquires is equal to force multiplied by the distance over which the force pushes the body. Consider the relative distances that the sledge and anvil move during impact. By picturing the impact in your mind, note that the sledge moves a greater distance while the anvil only ...

moves a smaller distance.* Equal force but unequal distances result in unequal changes in kinetic energy -- the sledge loses more kinetic energy than the anvil gains.

So while all the momentum of the sledge is transferred to the anvil and then to the professor, all the kinetic energy is not. The professor is shielded from the kinetic energy and he will return to lecture again.

* Another way to see this is to reason that during the impact, the sledge's speed drops from about 30 mph to 1 mph, while the anvil's speed increases from 0 mph to 1 mph. Althouh they both end up with the same speed, the sledge was moving faster than the anvil at all other instants and therefore had to move farther during the impact.

Answer 1

The hammer's mass m times the velocity of swing V is it's momentum p = mV. That is transferred to the anvil and the fearless physics professor of total mass M (fearless and physics prof. are redundant you know). Therefore, the anvil and prof will have a momentum of P = Mv after the blow. From the conservation of momentum p = mV = Mv = P; so that v = (m/M)V. Thus, a) is incorrect because momentum is not diminished by the anvil-prof mass combination. What about kinetic energy?

KE = 1/2 mV^2 is the kinetic energy for the hammer and ke = 1/2 Mv^2 = 1/2 M(m/M)^2 V^2 = 1/2 mV^2 (m/M) = KE(m/M) is the kinetic energy of the anvil-prof mass. In which case, if the mass of the hammer is m < M and m/M < 1.00, we have ke < KE; so that the kinetic energy of the anvil-prof mass M combination is less than that of the hammer.

That is, the hammer's kinetic energy is reduced by the M of the anvil and prof combined. But because we do not know how much of M is the anvil and how much is the prof. it is unclear how much of the kinetic energy of the hammer is shielded by the anvil and how much is shielded (diminished) by the prof herself. The answer is b) with the caveat just cited.

c) is incorrect because momentum is not shielded. d) is incorrect because kinetic energy is.