Question

Prove Work-Energy Theorem For A Constant Force.

Answer 1

Step 1: Given

Work done in moving the body is $W$

Change in kinetic energy of the body is $∆K$

Force experienced by the body is $F$.

Step 2: Formula Used

Work done by a body is $W=Fx$, where $F$ is the force experienced by the body and $x$ is the displacement of the body.

Force acting on a body is $F=ma$, where $m$ is mass of body and $a$ is acceleration of the body.

Newton's third equation of motion is $v^2=u^2+2ax$, where $v$ is the final velocity, $u$ is the initial velocity, $a$ is the acceleration and $x$ is the displacement.

Kinetic energy of a body is $K=\frac{1}{2}m{v}^2$, where $m$ is the mass of the body and $v$ is the velocity.

Step 3: Prove the work-energy theorem for a constant force.

For a force acting on a body of mass $m$, the work done will be

$\begin{array}{rcl}W& =& Fx\\ & =& \left(ma\right)x\end{array}$

Find the final velocity of the body which is initially at rest using Newton's third equation of motion.

$\begin{array}{rcl}{v}^2& =& u^2+2ax\\ & =& 0+2ax\\ & =& 2ax\end{array}$

Find the change in kinetic energy of the body

$\begin{array}{rcl}∆K& =& \frac{1}{2}m{v}^2\\ & =& \frac{1}{2}m\left(2ax\right)\\ & =& max\end{array}$

Hence, Work done is equal to change in Kinetic energy.