Try it on paper first. A hint, the key idea and the worked solution are here when you need them.
A block of mass on a smooth horizontal table is joined to a light spring of force constant , whose other end is fixed. When the block is at rest, its right face is from a rigid wall, as in the figure. The block is pulled to the left of its rest position and released. Every time it reaches the wall it bounces back elastically, and the bounce takes no time.
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Plan. Use the ordinary motion up to the wall. The bounce makes the cycle symmetric, so the time between hits is twice the time from release to the wall.
Step 1. Speed at the wall (a). . At :
Step 2. Time between hits (b). From release, first holds at , so . After the bounce the block retraces the motion back to and returns: the time between hits is . Without the wall the period would be .
Step 3. Average force (c). Each hit gives the wall . So the average force is
Step 4. Time to the right (d). The block is to the right of its rest position from to the wall and back. Going out takes , so the time on that side is per cycle, a fraction .
Answer: (a) . (b) . (c) . (d) .
Check. Over a cycle the block's momentum comes back to its value, so the wall's average push must balance the spring's average pull. The block spends most of its time on the left, where the spring pushes it to the right, so on average the wall must push it to the left.
Watch out. Halving the free period, , is wrong. The wall cuts the cycle where the block is fast, so the time saved is less than half the cycle.
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