Block B has a mass of 50kg & rests on the smooth surface of Cart C having a mass

Question

Block B has a mass of 50kg & rests on the smooth surface of Cart C having a mass of 75kg. A massless spring with a spring constant k=300N/m is attached to the cart. The block B is pushed against the spring, compressing it by 0.2m. System is released from rest. Find the relative velocity of the block with respect to the cart when the spring reaches its unstretched position. Neglect the mass of the cart's wheels & also friction between the cart & the ground.

Explanation / Answer

let the velocity w.r.t to ground be

vB to the right

and vC to the left

then conserving momentum,

50vB = 75vC   => vB = 1.5vC

conserving energy

0.5 x 300 x (0.2)2 = 0.5 x 50 x (vB)2 + 0.5 x 75 x (vC)2

12 = 50 (1.5vC)2 + 75 (vC)2

vC = 0.253 m/s => vB = 1.5 x 0.253 = 0.379 m/s

relative velocity = vB + vC = 0.253 + 0.379 = 0.632 m/s

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