A worker drives a 0.75 kg spike into a rail tie with a 3.0 kg hammer. The hammer hits the spike w/ a speed of 73 m/s. If 43% of the hammer's. . .
kinetic energy is converted into the internal energy of the spike, how much does the internal energy of the spike increase?
Here we have a significant concept: the conversion of one form of energy to another. In this case, we are converting kinetic energy of the hammer to the "internal energy" of the spike.
I will say now that "internal energy" has varying definitions depending on with whom you talk, and the way it's presented here doesn't lend much to our figuring out what the heck the problem is talking about. The thermodynamic definition (the only official definition I can think of) provided in the first link below basically is science-speak for "whatever energy we can't easily measure directly in a system." However, the best part for us is that it doesn't matter what the problem means by "internal energy"!
Instead, we'll take a conservation of energy approach. We'll assume that initially, the only energy present is the kinetic energy of the hammer and the initial internal energy of the spike. In the end, we'll assume that any remaining kinetic energy in the hammer is retained as kinetic energy in the hammer. That may have been hard to follow...here's another way of putting it given our assumptions:
K(i) =Initial Kinetic Energy of Hammer
K(f) = Final Kinetic Energy of Hammer
I(i) = Initial Internal Energy of spike
I(f) = Final Internal Energy of spike):
K(i) + I(i) = K(f) + I(f)
The problem gives us some numbers to work with. We know that 43% of the K.E. of the Hammer goes into the I.E. of the Spike. We also can assume no energy is transferred from the spike to the hammer, and we'll just go ahead and assume the hammer retains the rest of its K.E., which is 57% of it's starting K.E.
Now, given our assumptions, we can revamp the above equation:
K(i) + I(i) = 0.57K(i) + (0.43K(i) + I(i))
The term in parentheses in the last part of the equation will be the final internal energy of the spike. To find the increase in energy, we will subtract the final internal energy from the initial internal energy:
Increase in internal energy = I(f) - I(i)
Substituting the above value for the final internal energy of the spike:
Increase in internal energy = 0.43K(i) + I(i) - I(i)
Increase in internal energy = 0.43K(i)
Alright, in order to calculate the increase in internal energy, we'll need to find the initial kinetic energy of the hammer. To find kinetic energy, we simply use the following equation:
Kinetic Energy = 1/2*m*v^2
where m = mass of object, v = velocity of object
In our case, the mass of the hammer is 3.0 kg and the initial velocity of the hammer is 73 m/s, giving us the following initial kinetic energy (remember K(i) is the initial kinetic energy of the hammer):
K(i) = 1/2*(3.0 kg)*(73 m/s)^2 = 7993.5 J
Using our above relation for the increase in internal energy, we can now calculate its exact value using this initial value for the kinetic energy of the hammer:
Increase in Internal Energy = 0.43K(i)
= 0.43(7993.5 J) = 3437.2 J
And there you have your answer. You'll notice that we didn't need to assume that the rest of the kinetic energy of the hammer remained in the hammer. Pretty much that other 57% of the hammer's kinetic energy doesn't matter for this problem. Notice, too, that the fact that the mass of the spike had no bearing on the problem, either!
All we needed to know was the kinetic energy of the hammer and the fact that 43% of it was converted to "internal energy." I feel that learning to filter out the information that doesn't matter is as important a take-away point from this problem as the correct answer.