Work and Kinetic Energy

Kinetic energy If a variable force of magnitude $F(x)$ moves

an object of mass $m$ along the $x-$axis from $x_1$ to $x_2,$ the object's

velocity $v$ can be written as $d\frac{x}{d}t($where $t$ represents time$).$ Use

Newton's second law of motion $F=m(d\frac{v}{d}t)$ and the Chain

Rule

$\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}=v\frac{dv}{dx}$

to show that the net work done by the force in moving the object

from $x_1$ to $x_2$ is

$W={\displaystyle \underset{x_1}{\overset{x_2}{}}}F(x)dx=\frac{1}{2}m{v}_2^2-\frac{1}{2}m{v}_1^2,$

where $v_1$ and $v_2$ are the object's velocities at $x_1$ and $x_2.$ In physics,

the expression $(\frac{1}{2})m{v}^2$ is called the kinetic energy of an object

of mass $m$ moving with velocity $v.$ Therefore, the work done by

the force equals the change in the object's kinetic energy, and we

can find the work by calculating this change.