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

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?