A box of mass m is pressed against (but is not attached to) an ideal spring of f

Question

A box of mass m is pressed against (but is not attached to) an ideal spring of force constant k and negligible mass, compressing the spring a distance x. After it is released, the box slides up a frictionless incline as shown in the figure and eventually stops. If we repeat this experiment but instead compress the spring a distance of 2x just as it moves free of the spring, the box will have twice as much kinetic energy as before. just before it is released, the box has twice as much elastic potential energy as before. just as it moves free of the spring, the box will be traveling twice as fast as before. just as it moves free of the spring, the box will be traveling four times as fast as before. the box will go up the incline twice as high as before.

Explanation / Answer

here,

initial elastic potential energy of the spring , PE = 0.5 * k * x^2

when x' = 2 * x

PE' = 0.5 * k * 4 * x^2

PE' = 4 * PE

using conervation of energy

KE' = 4 * KE

when the kinetic is four times,

then the velocity is doubled

so,

when the spring is compressed2 x , just as it moves free of the spring , the box will be travelling twice as fast as before

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