Randall D. Knight — Dynamics I: Motion Along a Line: Problem 75

A spring-loaded toy gun exerts a variable force on a plastic ball as the spring expands. Consider a horizontal spring and a ball of mass \( m \) whose position when barely touching a fully expanded spring is \( x = 0 \). The ball is pushed to the left, compressing the spring. You’ll learn in Chapter 9 that the spring force on the ball, when the ball is at position \( x \) (which is negative), can be written as \((F_{Sp})_x = -kx\), where \( k \) is called the spring constant. The minus sign is needed to make the x-component of the force positive. Suppose the ball is initially pushed to \( x_0 = -L \), then released and shot to the right. a. Use what you’ve learned in calculus to prove that \( a_x = v_x \frac{dv_x}{dx} \) b. Find an expression, in terms of \( m \), \( k \), and \( L \), for the speed of the ball as it comes off the spring at \( x = 0 \).