A spring has an unstretched length of 032 m A block with mas

A spring has an unstretched length of 0.32 m. A block with mass 0.2 kg is hung at rest from the spring and the spring becomes 0.4 m long. Next, the spring is stretched to a length of 0.43 m and the block is released from rest. Air resistance is negligible.

(a) How long does it take for the block to return to where it was released?

(b) Next, the block is again positioned at rest, hanging from the spring (0.4 m long). A bullet of mass 0.003 kg traveling at a speed of 200 m/s straight upward buries itself in the block, which then reaches a maximum height h above its original position. What is the speed of the block immediately after the bullet hits?

(c) Now write an equation that could be used to determine how high the block goes after being hit by the bullet (a height h), but you need not actually solve for h.

Solution

m g = k (L₁ - L₀)

0.2 * 9.81 = k (0.4 - 0.32)

k = 24.52 N/m

Time for the block to return to where it was released

= 2 pi sqrt[ m/k] = 2 pi * sqrt [0.2/24.52]

= 0.567 s

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b) From conservation of momentum

(0.2 * 0 ) + (0.003 * 200) = (0.2 + 0.003) * v

v = 2.96 m/s