A spring of constant k is attached to a block of mass m on a smooth surface. Initially the spring is at natural length x=0; it is compressed to x=-a and released. The coefficient of restitution is e=1/2. If the velocity just after the first collision is $a\sqrt{\frac{nk}{16m}}$ , find n.
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Question

A spring of constant k is attached to a block of mass m on a smooth surface. Initially the spring is at natural length x=0; it is compressed to x=-a and released. The coefficient of restitution is e=1/2. If the velocity just after the first collision is $a\sqrt{\frac{nk}{16m}}$ , find n.

Your answer

G1 Physics · Accepted Answer

The block starts at x=-a with zero speed. Before hitting the fixed wall at x=a/2, energy conservation gives speed before collision=ω√(a²-a²/4)=aω√3/2, where ω=√(k/m). With coefficient of restitution e=1/2 against a fixed wall, speed just after collision is speed after collision=(1/2)speed before collision=aω√ 3 4 \frac{3}{4} 4 3 ​ =a√(3k/16m). Comparing with a√(nk/16m) gives n=3.