# The force applied against an object moving horizontally on a linear path is described by F(x)=x^2+ x-1 . By how much does the object's kinetic energy change as the object moves from  x in [ 1, 2 ]?

Dec 28, 2015

$2.5 \text{J}$

#### Explanation:

Work done = force x distance moved in the direction of the force.

Because the force applied is a function of distance we get:

$W = {\int}_{1}^{2} F . \mathrm{dx}$

$\therefore W = {\int}_{1}^{2} \left({x}^{2} + x - 1\right) . \mathrm{dx}$

$W = {\left[{x}^{3} / 3 + {x}^{2} / 2 - x\right]}_{1}^{2}$

$W = \left[\frac{8}{3} + \frac{4}{2} - 2\right] - \left[\frac{1}{3} + \frac{1}{2} - 1\right]$

$W = \left[2.66 + 2 - 2\right] - \left[\frac{1}{3} + \frac{1}{2} - 1\right]$

$W = 2.66 - 0.166$

$W = 2.5 \text{J}$

This work will appear as an increase in kinetic energy.