Question

How work and energy are related?

Answer 1

Work and energy,

1. Work is defined as the product of applied force and the body's displacement. When a force displaced an object, work is said to be done. The amount of energy that is required to displace a body from one point to another point, is called work done.
2. Work done, $W=F.ds$ where $F$ is the applied force and $ds$ is the displacement.
3. Energy is the ability for doing work.
4. Energy and work both are scalar quantities.

Relation between work and energy,

If a body is moving from $b$ to $a$, with a velocity $v$ and the displacement of the body is $s$. then the work done by the body is given by,

Work done,

$$\begin{gathered} W={\int }_b^{a}F.ds \\ orW={\int }_b^{a}m\frac{dv}{dt}.ds(\because F=m\frac{dv}{dt}=appliedforce) \\ orW=m{\int }_b^a\frac{ds}{dt}.dv \\ orW=m{\int }_b^{a}v.dv(\because \frac{ds}{dt}=v=velocity) \end{gathered}$$.

⇒ $W=\frac{1}{2}m{{\mathrm{v}}_{\mathrm{a}}}^2-\frac{1}{2}m{{\mathrm{v}}^2}_{\mathrm{b}}.$

Where the term $\frac{1}{2}m{v^2}_a-\frac{1}{2}m{v_b}^2$indicates a change in kinetic energies.

Thus, work done = change in kinetic energy.