# The force applied against an object moving horizontally on a linear path is described by F(x)= 4x^2+x . By how much does the object's kinetic energy change as the object moves from  x in [ 0, 4]?

Apr 9, 2018

By the work energy theorem, change in kinetic energy is the work done by the object.

It is given that, $F \left(x\right) = 4 {x}^{2} + x$

and $W = F . x$ $\textcolor{w h i t e}{p p w w w p p w w w p p}$ $\left[\text{let x be displacement}\right]$

$\mathrm{dW} = F . \mathrm{dx}$

Integrating this,

$\int \mathrm{dW} = \int F . \mathrm{dx}$

$\int \mathrm{dW} = {\int}_{0}^{4} d \left(4 {x}^{2} + x\right) . \mathrm{dx}$

$W = {\left[4 {x}^{3} / 3 + {x}^{2} / 2\right]}_{0}^{4}$

$W = 4 {\left(4\right)}^{3} / 3 + {\left(4\right)}^{2} / 2$

$W = 4 {\left(4\right)}^{3} / 3 + {\left(4\right)}^{2} / 2$

$W = \frac{256}{3} + 8 = \frac{256}{3} + \frac{24}{3} = \frac{280}{3} = 93.3 N s$