Item 11 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. The spring force on the ball, when the ball is at position x (which is negative), can be written as (FSp)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.

Part A

Use what you've learned in calculus to write the correct proof so that a_{x}= v_{x}\frac{dv_{}{_{x}}}{dx}

Part B

Find an expression, in terms of m, k, and L, for the speed of the ball as it comes off the spring x = 0.