Randall D. Knight — Interactions and Potential Energy: Problem 62

A clever engineer designs a “sprong” that obeys the force law \(F_x = -q(x - x_{eq})^3\), where \(x_{eq}\) is the equilibrium position of the end of the sprong and \(q\) is the sprong constant. For simplicity, we’ll let \(x_{eq} = 0 \text{ m}\). Then \(F_x = -qx^3\). (a) What are the units of \(q\)? (b) Find an expression for the potential energy of a stretched or compressed sprong. (c) A sprong-loaded toy gun shoots a \(20 \text{ g}\) plastic ball. What is the launch speed if the sprong constant is \(40,000\), with the units you found in part a, and the sprong is compressed \(10 \text{ cm}\)? Assume the barrel is frictionless.