1. (25 pts)
a) (5 pts) Supposing you wanted to analyze the motion of a whole bunch of objects. What could you use
as the “position” of this system?
You could use the center of mass as the “position” of the system.
b) (5 pts) If two objects start from rest at the same elevation and then follow different paths (with no
friction) which end together (at the same spot), what can you say about the relative speeds of the two
objects when they pass the ending location?
Since there are no external/non-conservative forces, energy is conserved. The change in gravitational
potential energy is the same for both paths since they both have the same starting and ending points.
Therefore the difference in kinetic energy is the same for objects traveling down both paths thus mak-
ing the final velocities equal. Also note that since both kinetic energy and gravitational potential
energy depend on the mass of the object, the object mass has no effect on the final speed.
c) (5 pts) Explain why friction is not a conservative force.
There are two tests for a conservative force. Both must be met. If either fails, then it is not a conser-
vative force.
1. Work done around a closed loop must be zero. Work done by friction is not zero when you move
an object and return it to the starting location. Thus friction fails the first criteria.
2. Work done when moving the object between two points must not depend on the path between the
two points. Work done by friction depends on the distance moved, and thus on the path between
the two points. So friction fails the second criteria also.
Just saying that friction takes energy out of the system is not enough to classify it as non-conservative.
Depending on how you define your system, gravity can also take energy out of it but is still a conserva-
tive force. And whether or not the force of friction is constant does not matter. Both gravitational and
spring forces are not constant, yet are conservative.
Also note that we need to figure out the work done by friction, not the frictional force itself. Force does
not equal work!
d) (5 pts) You find yourself alone in a small boat with nothing else but a large bag full of grape fruit.
The water is calm, there is no wind, and you’re not that far from shore. How could you at least get the
boat started moving (without trying to paddle it with your hands).
You could start throwing grape fruit as hard as you can out the back of the boat. Since the mass of
grape fruit is moving to the back, the mass of you and the boat must move in the opposite direction in
order to conserve momentum (and to keep the center of mass at the same spot).
Sure, you could walk from one end of the boat to the other, or even throw the fruit to the other end of
the boat. The boat will move while you are moving. But when you stop moving, or the fruit lands in the
boat, the boat will again stop moving and you’re back in the same situation as before: a boat at rest in
the water.
e) (5 pts) Is momentum conserved when an object is in free fall? Explain.
It is not conserved. Why? There is a non-zero net force (gravity). Also note that by our definition of
“free fall,” there is no friction in this case.
I suppose it could be argued that momentum is conserved if you include both the falling object and
the earth together as a system. Then gravity is an internal force and won’t change the momentum of
the system. So as the falling object picks up speed, the earth must also accelerate upward to keep the
net momentum of the system zero. But in order to get credit for this, you’d have to state that you were
including the earth in your system and that it is the momentum of the system that is conserved, not
the momentum of the falling object by itself.
The fact that gravity is a conservative force has nothing to do with whether or not momentum is
conserved.
