Serway & Jewett — Energy of a System: Problem 41.

You have a new internship, where you are helping to design a new freight yard for the train station in your city. There will be a number of dead-end sidings where single cars can be stored until they are needed. To keep the cars from running off the tracks at the end of the siding, you have designed a combination of two coiled springs as illustrated in Figure P7.41. When a car moves to the right in the figure and strikes the springs, they exert a force to the left on the car to slow it down. Both springs are described by Hooke’s law and have spring constants \(k_1 = 1600 \text{ N/m}\) and \(k_2 = 3400 \text{ N/m}\). After the first spring compresses by a distance of \(d = 30.0 \text{ cm}\), the second spring acts with the first to increase the force to the left on the car in Figure P7.41. When the spring with spring constant \(k_2\) compresses by \(50.0 \text{ cm}\), the coils of both springs are pressed together, so that the springs can no longer compress. A typical car on the siding has a mass of \(6000 \text{ kg}\). When you present your design to your supervisor, he asks you for the maximum speed that a car can have and be stopped by your device.