Chapter 6, Problem 74EAP

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 x₀= -L, then released and shot to the right.

a. Use what you've learned in calculus to prove that

$a_x=v_x\frac{d{v}_x}{d_x}$

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.