A box A is at rest on a frictionless horizontal surface and connected to a wall

Question

A box A is at rest on a frictionless horizontal surface and connected to a wall by an ideal spring with an unknown spring constant as shown in the figure. A box B is moving to the left at the speed V_1. The two boxes then collide and stick together. The string then is compressed by the motion of this two boxes. The maximum compression of the spring is x_1. This experiment is repeated with the speed V_2 of the box B. What is the new maximum compression of the spring in terms of V_1, V_2, and x_1?

Explanation / Answer

if initial velocity is v
Then for collision, use conservation of momentum,
m1*v1i = (m1+m2)*vf
m*v = 2m*vf
vf = v/2

For compression use conservation of energy
0.5*(m+m)*vf^2 = 0.5*k*x^2
x^2 = (2m)*v/2k
        = m/vk

for case 1:
x1^2 = m/(V1*k)
x2^2 = m/(V2*k)
divide one by other:
x2^2/x1^2 = V1/V2
x2/x1 = sqrt (V1/V2)
x2 = x1* sqrt (V1/V2)

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