# Question #32114

Dec 25, 2016

According to conservation of energy, the final velocity will be

${v}_{f} = \sqrt{{v}_{i}^{2} + 2 g h - \frac{2 f l}{m}}$

#### Explanation:

This problem is best done using conservation of energy methods.

There will be three terms in the expression, as there are three different energy changes that occur - kinetic, potential and frictional heating (shown in that order in the equations below).

$\Delta K + \Delta U + \Delta E = 0$

Using the symbols given in the problem

$\frac{1}{2} m {v}_{f}^{2} - \frac{1}{2} m {v}_{i}^{2} - m g h + f l = 0$

where the middle term for the potential energy is subtracted because this will be an energy decrease.

We now rearrange the make ${v}_{f}$ the subject of the expression

$\frac{1}{2} m {v}_{f}^{2} = \frac{1}{2} m {v}_{i}^{2} + m g h - f l$

Multiply each term by 2 and divide each by $m$

${v}_{f}^{2} = {v}_{i}^{2} + 2 g h - \frac{2 f l}{m}$

and finally

${v}_{f} = \sqrt{{v}_{i}^{2} + 2 g h - \frac{2 f l}{m}}$