*Answer:* Referring to the diagram, the force exerted by the
pirate can be resolved into a horizontal component and a vertical
component . Since the chest only moves horizontally, the vertical
component of performs zero work. The work performed by the horizontal
component is simply the magnitude of this component times the horizontal distance
moved by the chest:

The chest is subject to the following forces in the vertical direction:
the downward force due to gravity, the upward reaction force
due to the dock, and the upward component of the force
exerted by the pirate. Since the chest does not accelerate in the
vertical direction, these forces must balance. Hence,

The frictional force is the product of the coefficient of kinetic friction and the normal reaction , so

The work done by the frictional force is

Note that there is a minus sign in front of the because the displacement of the chest is in the opposite direction to the frictional force. The fact that is negative indicates a loss of energy by the chest: this energy is dissipated as heat via friction. Hence, the dissipated energy is .

The final kinetic energy of the chest (assuming that it
is initially at rest) is the difference between the work
done by the pirate and the energy dissipated as heat.
Hence,

Since , the final velocity of the chest is