Two objects, X and Y, are held at rest on a horizontal frictionless surface and a spring is compressed between them. The mass of X is 2/5 times the mass of Y. Immediately after the spring is released, X has a kinectic energy of 50 J and Y has a kinetic erengy of:
A) 20 J
B) 8 J
C) 310 J
D) 125 J
E) 50 J

Question

Quizplus · Accepted Answer

According to the law of conservation of energy, the total mechanical energy of the system (spring + X + Y) is conserved. Thus, the initial potential energy of the spring is equal to the sum of the final kinetic energies of X and Y. Let the initial compression of the spring be x and the spring constant be k. Then, the initial potential energy of the spring is (1/2)kx^2. When the spring is released, X and Y move in opposite directions with equal speeds because of the conservation of momentum. Let the speed of each object be v. Then, using the conservation of momentum, we can write:
(mass of X)(v) = (mass of Y)(-v)
Solving for v, we get v = sqrt[(mass of Y)/(mass of X)](sqrt(50 J))
Therefore, the final kinetic energy of X is (1/2)(mass of X)(v^2) = 125 J.
Since the total final kinetic energy must be equal to the initial potential energy of the spring, the final kinetic energy of Y is 125 J - 50 J = 75 J. However, the question asks for the kinetic energy of Y immediately after the spring is released, which is when the spring is at its maximum compression and X has not yet started to move. At this point, all the initial potential energy of the spring is in the form of elastic potential energy, and none of it has been converted to kinetic energy. Therefore, the kinetic energy of Y immediately after the spring is released is 0 J.

Two Objects, X and Y, Are Held at Rest on a Horizontal