# A ball with a mass of 200 g is projected vertically by a spring loaded contraption. The spring in the contraption has a spring constant of 54  kgs^-2 and was compressed by 7/4 m when the ball was released. How high will the ball go?

Jun 4, 2016

The spring potential energy of the spring will be converted into kinetic energy as the ball flies upward, then into gravitational potential energy as it reaches the top of its flight. Its highest point will be at $42.2$ $m$.

#### Explanation:

The spring potential energy is given by:

${E}_{p} = \frac{1}{2} k {x}^{2}$

Where $k$ is the spring constant and $x$ is how far it has been compressed:

${E}_{p} = \frac{1}{2} \times 54 \times {\left(\frac{7}{4}\right)}^{2} = 82.7$ $J$

We need to express measurements in SI units, so the $200$ $g$ mass is $0.2$ $k g$.

We don't need to calculate the kinetic energy, we can move straight to the gravitational potential, which will be equal to the spring potential energy:

${E}_{k} = m g h$

Rearranging:

$h = {E}_{k} / \left(m g\right) = \frac{82.7}{0.2 \times 9.8} = 42.2$ $m$