# Pushing a ball through a viscous fluid

We push a ball through a viscous fluid with constant external force. As the ball moves, it compresses a spring. The spring resists compression with an elastic force f=kd, where k is the spring constant. When this force balances the external force, the ball stops moving at d=f/k. Throughout the process, the applied force is fixed, so the work done is fd=f^2/k and energy stored in the spring is 1/2kd^2 or 1/2f^2/k.
Suppose that we suddenly reduce the external force to a value of f1 that is smaller than the original external force.The ball moves in the opposite direction.
a. How far does the ball move and how much work it does against the external force f1?
b. For what constant value of f1 will the useful work be maximal? Show that the useful work output is half of what is stored in the spring =1/4f^2/k.
c. How could we make this process more efficient?

a. The elastic force is equal to the external force + friction force? How do we get the distance? I am confused.
b. The work that is done on the ball by the spring is 1/4f^2/k. Do we need to include the friction force here when the ball is now moving in the opposite direction?
c. Free energy transduction is most efficient when it proceeds by the incremental, controlled release of many small constraints. What steps do we need to take to make it more efficient? What are the constraints?