Homework #5
Conceptual Solution

1. (a). To calculate the velocity of the center of mass of the child/sled system, we use conservation of momentum (conserved because no net external force acts on the child/sled system). Thus

   

   where v' is the final velocity of both child and sled after the collision (thus also the final velocity of their center of mass). The initial sled velocity is zero, so

   

2. (b). Increasing either the child's initial velocity or mass increases the child/sled system's initial momentum, thus increasing the final center of mass velocity v' of the system (equal to the child's final velocity). Decreasing the child's mass reduces the system's momentum and hence reduces its final velocity. Increasing the sled's mass increases the child/sled system's mass, thus decreasing its final velocity at fixed momentum.
3. (b). Momentum is always conserved for collisions within an isolated system (where no net external forces act on the system). Kinetic energy is not conserved if the collision is inelastic. The example given here, where the child and sled stick together and move off with common final velocity, is the example of the most inelastic collision possible.
4. (d). By Newton's third law, when objects are in contact, they exert forces on each other that are equal in magnitude but opposite in direction. Thus (d) is correct and (a) incorrect. (b) is incorrect, because a system's momentum is conserved only when there are no external forces; here the log applies an external force to the child/sled system. Of course, noting that the child/sled system stopped, reducing its momentum and kinetic energy to zero, also shows that neither its momentum nor its kinetic energy were conserved. Thus (c) is incorrect as well.
5. (b) Note that, on the frictionless ice, the child/sled system is isolated -- no net external forces act on it, so its momentum is conserved. Thus, when the child tries to walk forward, the sled must slip backward, so that child plus sled continue to have zero momentum. (One can also see this from a center of mass argument -- the center of mass must retain its constant (zero) velocity, and so remain fixed in place, so that when the girl moves forward, the sled moves backward to keep the center of mass stationary.) The girl's effort in walking forward thus produces forward motion relative to the backward sliding sled. Relative to the fixed log, she moves forward more slowly than if she were producing this forward motion relative to the fixed ground.
6. (c). Again, the key here is that the child/sled system is isolated, with constant zero momentum and fixed center of mass. Walking to the back of the sled and sitting down will cause the sled to slide a bit forward (constant center of mass) then stop (final momentum remains zero). Walking back and forth rapidly will make the sled also rock back and forth, opposite her direction of motion, but she and the sled will remain stuck with their center of mass at the same position -- they can make no net progress back to the edge of the lake. However, by throwing her oversized (massive) hat away from the sled, the child can impart momentum to the system of the sled plus her hatless self. Because child plus hat plus sled comprise an isolated system, with zero initial momentum, tossing the hat away with momentum p means that the child and sled together must acquire momentum p in the opposite direction, enabling them to glide back to their starting point.
7. (b) Since you are told that the child/sled system slows down in this case, it is clear simply by definition ( ) that neither momentum nor kinetic energy are conserved. This makes sense because it is the action of an external force (friction) that slows down the sled. Due to the external force, momentum is not conserved. Because the external force is nonconservative (friction), mechanical energy is not conserved either, and (b) is the only correct choice.

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