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On a frictionless surface two blocks of masses m and M move in the same direction with speeds v1 and v2 (v1 > v2), with M ahead of m. An ideal spring of force constant K is fixed to the rear face of M. Find the maximum compression of the spring when the blocks interact.

1. v1 * sqrt(m/K)
2. v2 * sqrt(M/K)
3. (v1 - v2) * sqrt(mM/((M + m)K))
4. None of the above.

Correct answer: (v1 - v2) * sqrt(mM/((M + m)K))

Solution

At maximum compression both blocks share a common velocity; the lost kinetic energy equals the spring PE. Using reduced mass mu = mM/(M+m), (1/2)K x² = (1/2) mu (v1 - v2)², giving x = (v1 - v2) sqrt(mM/((M+m)K)).

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