Chapter 8, Problem 30P

A crate on rollers is being pushed without frictional loss of energy across the floor of a freight car (see the following figure). The car is moving to the right with a constant speed $v_0$ . If the crate starts at rest relative to the freight car, then from the work-energy theorem, $Fd=m{v}^2/2,$ where d, the distance the crate moves, and $v$ , the speed of the crate, are both measured relative to the freight car. (a) To an observer at rest beside the tracks, what distance $d^{\prime }$ is the crate pushed when it moves the distance d in the car? (b) What are the crate’s initial and final speeds $v_0{}^{\prime }$ and $v^{\prime }$ as measured by the observer beside the tracks? (c) Show that $Fd=m{(v^{\prime })}^2/2-m{\left(v^{\prime }{ }_0\right)}^2/2$ and, consequently, that work is equal to the change in kinetic energy in both reference systems.