The force applied against an object moving horizontally on a linear path is described by F(x)= sinpix +2 . By how much does the object's kinetic energy change as the object moves from  x in [ 2, 5 ]?

Jun 30, 2016

$= \frac{2}{\pi} + 6$

Explanation:

the change in KE will equal the work done assuming as seems OK to do here, that all work is converted into the object's motion, ie there are no losses to friction etc,

on that basis we can say that Work = Force X Distance

and also that Work = $\Delta K E$

or in maths form, bearing in mind we are working in only 1 dimension, that:

$\Delta K E = W = {\int}_{x 1}^{x 2} \mathrm{dx} q \quad F \left(x\right)$

so here we have

$\Delta K E = {\int}_{2}^{5} \mathrm{dx} q \quad \sin \pi x + 2$

$= {\left[- \frac{1}{\pi} \cos \pi x + 2 x\right]}_{2}^{5}$

$= \left[- \frac{1}{\pi} \cos 5 \pi + 10\right] - \left[- \frac{1}{\pi} \cos 2 \pi + 4\right]$

$= \left[- \frac{1}{\pi} \left(- 1\right) + 10\right] - \left[- \frac{1}{\pi} + 4\right]$

$= \frac{2}{\pi} + 6$