Elastic Potential Energy

Question

A .15 kg ball is compressing a vertical spring by .085 meters. The spring constant is 124n/m. What height will the ball reach?

Answer 1

This is an example of conservation of energy.

The spring energy before the release gets converted to potential energy

when the ball reaches maximum height.

Let

m = 0.15 kg me the mass of the ball,

d = 0.085 m be the compression of the spring,

k = 124 N/m be the spring constant,

g = 9.81 m/s² be gravitational acceleration,

h (to be computed) be the ball's maximum height above the original position.

First we simplify the problem by eliminating kinetic energy.

The statement of the problem does not say so explicitly, but I assume that

the ball is stationary before the release.

That makes the ball's kinetic energy zero before the release.

And it is also zero when the ball reaches the maximum height.

So kinetic energy is not involved in the energy exchange.

Now let's discuss potential gravitational energy.

That is always defined with respect to some (arbitrarily chosen) base level.

Let's chose the ball's original position as the base level.

That makes the initial potential energy 0, and the final potential energy

mgh

Now let's discuss the spring energy.

Before the release the spring stores energy

kd²/2

Afterward the spring can be assumed to return to its equilibrium position with zero energy.

The result is that we have only two non-zero energy levels:

the initial spring energy, and the final potential energy of the ball.

By conservation of energy

kd²/2 = mgh

From that

h = kd²/2mg

Substituting actual numbers

h = 124×0.085²/(2×0.15×9.81) ≈ 0.3 m