Question

How are work and kinetic energy related?

Answer 1

Hint: Work is normally said to be equal to the change in kinetic energy, but for showing the relation between work and kinetic energy it’s better to go from normal formula of work and then substituting the values in it will give us the value of work, which when compared with the formula of kinetic energy will give us the relationship between work and kinetic energy.
Formula used:
$\begin{align}
  & W=Fr \\
 & K.E.=\dfrac{1}{2}m{{v}^{2}} \\
 & {{v}^{2}}=v_{0}^{2}+2a(r-{{r}_{0}}) \\
\end{align}$

Complete answer:
We know that work is defined as the energy, you add to an object by applying force over some distance. So, the formula for force can be written as:
$W=Fr$,
where, ‘F’ is force, and
‘r’ is distance.
We know that the formula for kinetic energy is given as:
$K.E.=\dfrac{1}{2}m{{v}^{2}}$
Now, using one of the motion equations we can say that:
${{v}^{2}}=v_{0}^{2}+2a(r-{{r}_{0}})$
Here, the initial velocity ${{v}_{0}}$ and initial distance traveled ${{r}_{0}}$ are both zero, then
$\begin{align}
  & \Rightarrow {{v}^{2}}=0+2a(r-0) \\
 & \Rightarrow {{v}^{2}}=2ar \\
 & \Rightarrow r=\dfrac{{{v}^{2}}}{2a} \\
\end{align}$
And, from Newton’s second law we know that:
$F=ma$
Hence, Substituting the value of force and distance in the work equation, we get:
$\begin{align}
  & W=ma\times \dfrac{{{v}^{2}}}{2a} \\
 & \therefore W=\dfrac{1}{2}m{{v}^{2}} \\
\end{align}$
We know that,
 $K.E.=\dfrac{1}{2}m{{v}^{2}}$
$\therefore W=\Delta K.E.$, as work done is said to be equal to the change in kinetic energy.

Additional Information:
In physics, work represents a type of energy. Work is done when a force acts on something that undergoes a displacement from one position to another. Forces can vary as a function of position, and displacements can be along various paths between two points. We first define the increment of work done by a force acting through an infinitesimal displacement as the dot product of these two vectors. Then, we can add up the contributions for infinitesimal displacements, along a path between two points.
Kinetic energy is related to the forces acting on a body and was referred to as “the energy of motion.” The kinetic energy of a particle is one-half the product of the particle’s mass and the square of its speed.

Note:
This type of question is frequently asked in papers and the solution for it should be derived from the base equation so that proper explanation and relation can be put forward. If the solution is written directly without any steps of proof, then that solution is not considered to be a correct solution and the student might lose marks for the same.