Question

An object that is 0.5 m above the ground has the same amount of potential energy as a spring that is stretched 0.5 m. Each distance is then doubled.

How will the potential energies of the object and the spring compare after the distances are doubled?

The gravitational potential energy of the object will be two times greater than the elastic potential energy of the spring.
The elastic potential energy of the spring will be four times greater than the gravitational potential energy of the object.
The elastic potential energy of the spring will be two times greater than the gravitational potential energy of the object.
The potential energies will remain equal to one another.

Answer 1

The elastic potential energy will be two times greater.

Reason: gravity energy U_g = mgh scales linearly with height, so doubling h doubles U_g. Spring energy U_s = 1/2 k x^2 scales with x^2, so doubling x increases U_s by 4. Since they were equal at 0.5 m, after doubling U_s = 4U_initial and U_g = 2U_initial, so U_s is 2×U_g.