Conservation of mechanical energy

Mechanical energy is the sum of the kinetic and gravitational potential energy in the system. Since energy is never created of destroyed we know the mechanical energy must be constant for a given system. Using this we can calculate systems that were unsolvable with suvat's.

Example

A particle of mass $13\text{ kg}$ is fired up a smooth plane inclined at $25^{\circ}$ to the horizontal. Its initial speed was $16\text{ ms}^{-1}$, find the vertical distance it moved up the plane when its traveling at $10\text{ ms}^{-1}$.

$$\begin{align} \text{G.P.E gained} &= \text{K.E lossed}\\ &= \left(\frac{1}{2} \times 13 \times 16^2\right) - \left(\frac{1}{2} \times 13 \times 10^2\right)\\ mgh &= 1014\\ 13 \times 9.8 \times h &= 1014 \\ h &= 7.959\\ h &= 7.96\text{ m}\\ \end{align}$$