A spring-loaded toy gun exerts a variable force on a

Chapter 6, Problem 6.113

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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. Youll 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 1FSp2x = -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 youve learned in calculus to prove that ax = vx dvx 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