# The force applied against an object moving horizontally on a linear path is described by F(x)= cospix+x . By how much does the object's kinetic energy change as the object moves from  x in [ 0, 2 ]?

Apr 25, 2017

I got: $\Delta K = 2 J$

#### Explanation:

We can use the Work-K.E. theorem that tells us that:

$W = \Delta K$

or that work done on the system is equal to the change in kinetic energy.

Here we need to evaluate the work of a variable force so we use the integral version of work as:

$W = {\int}_{{x}_{1}}^{{x}_{2}} f \left(x\right) \mathrm{dx} = {\int}_{0}^{2} \left[\cos \left(\pi x\right) + x\right] \mathrm{dx} = {\int}_{0}^{2} \left[\cos \left(\pi x\right)\right] \mathrm{dx} + {\int}_{0}^{2} \left[x\right] \mathrm{dx} = \sin \frac{\pi x}{\pi} + {x}^{2} / s {|}_{0}^{2} = \sin \frac{2 \pi}{\pi} + {2}^{2} / 2 - 0 = 2 J$

So:

$W = \Delta K = 2 J$