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 v0. If the crate starts at rest relative to the freight car, then from the work-energy theorem, Fd = mv2 /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′ is the crate pushed when it moves the distance d in the car? (b) What are the crate’s initial and final speeds v0 ′ and v′ as measured by the observer beside the tracks? (c) Show that Fd′ = m(v′)2 /2 − m(v′0)2 /2 and, consequently, that work is equal to the change in kinetic energy in both reference systems.
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