A mass of 4.0 kg is in motion along a horizontal, frictionless surface with a speed of 2.5 m/s when it makes contact with a horizontal spring anchored to a wall. The spring compresses and brings the mass to a momentary rest. The spring constant is 1250 N/m. a) Determine the maximum distance the spring is compressed. b) Determine the speed of the block when the spring is compressed a distance of 9.5 cm.

Simply use conservation of energy

The initial energy is purely kinetic: 0.5mv2

When the spring is compressed to the max, the energy is purely potential: 0.5kx2

When the spring is partially compressed, the energy is a mixture of both: 0.5mv2+0.5kx2

Since the energy is a constant at every step of the way, you can equate the initial energy to the maximum potential energy and isolate x to find the maximum compression, then, for a known compression, you can equate the total energy to the third equation I wrote and isolate the final velocity

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