A 2 kg cart is held onto a spring (1,037 N/m) and is compressed 1.1 m. The cat is released and travels up a hill. At the top of the hill the cart is traveling at 4 m/s, how
much energy is converted to gravitational energy?

611.385 joules are converted to gravitational potential energy.

Explanation:

Let suppose that the cart begins at a height of 0 meters. The cart-spring system is conservative. By the Principle of Energy Conservation we have the following model:

(1)

Where:

, - Initial and final elastic potential energy, in joules.

, - Initial and final gravitational potential energy, in joules.

, - Initial and final translational kinetic energy, in joules.

And the energy converted to gravitational potential energy is calculated by (1) and definitions of translational kinetic and elastic potential energies:

(2)

Where:

- Spring constant, in newtons per meter.

- Initial compression of the spring, in meters.

- Mass, in kilograms.

- Final speed of the cart, in meters per second.

If we know that , , and , then the change in gravitational potential energy is:

611.385 joules are converted to gravitational potential energy.

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force required to detach it is given by

charges are taken to be fundamental charge

now we have

so above is the minimum distance at which it will detach