Conservation of Energy Problem.

[TwinGemini14]

A cart, initially at rest, slides down a frictionless ramp onto a horizontal frictionless surface which is a distance h below the original position of the cart. It then collides with the free end of a relaxed horizontal spring, the other end of which is fixed to a wall. As a result the spring compresses a distance D.

Suppose now the initial height is changed to 2h. How far will the spring now compress?

A) sqrt(2)D
B) 2D
C) 4D

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Initially: mgh = 0.5 k D^2

Suppose: mg(2h) = 0.5 k (2D)^2

So... the spring will now compress 4D. Is this correct?

 

[LowlyPion]

Maybe rework your math again?

And remember D is unknown.

 

[nlightner]

Your calculations are correct. The conservation of energy principle states that energy cannot be created or destroyed, only transferred between different forms. In this problem, the potential energy of the cart at the top of the ramp is converted into kinetic energy as it slides down the ramp. When it collides with the spring, the kinetic energy is converted into potential energy as the spring compresses. Since the initial height is doubled, the potential energy at the top of the ramp is also doubled, resulting in the spring compressing four times as much as before. This can be seen in the equation mgh = 0.5 k D^2, where both the mass and gravitational acceleration remain constant, but the height and compression distance are doubled. Therefore, the correct answer is C) 4D.