We are asked to compare the energy of a mass at the top of a ramp with the energy at the bottom:
The energy at the top is the gravitational potential energy and is given by:
[tex]U=mgh[/tex]Where:
[tex]\begin{gathered} m=\text{ mass} \\ g=\text{ acceleration of gravity} \\ h=\text{ height} \end{gathered}[/tex]The energy at the bottom is kinetic energy and is given by:
[tex]K=\frac{1}{2}mv^2[/tex]Where:
[tex]v=\text{ velocity}[/tex]Due to the principle of conservation of energy, the energy at the top is equal to the energy at the bottom is no energy is lost. Therefore, we have:
[tex]U=K[/tex]Or:
[tex]mgh=\frac{1}{2}mv^2[/tex]