Question

Suppose a particle is moving in one dimension from rest under the influence of a force that changes with the distance travelled by the particle as represented in the figure. Calculate the kinetic energy of the particle after it has travelled $3m$.

$\begin{align}
  & A.6.5J \\
 & B.2.5J \\
 & C.4J \\
 & D.5J \\
\end{align}$

Answer 1

Hint: The work energy theorem is the basis for solving the question. The work done by the force on a particle will be equivalent to the change in kinetic energy. The work done by the particle is given as the area under the force displacement graph. This all will help you in answering this question.

Complete step by step answer:
According to the work energy theorem, the work done by the force on a particle will be equivalent to the change in kinetic energy.
That is we can write that,
$\text{work done by force on particle=change in kinetic energy}$
As we all know, the work done by the particle can be found by taking the area under the force displacement graph.
That is,
Work done= area under the F-X graph
The area under the graph can be found using the equation,
$W=\int{F\cdot ds}$
From the graph we can write that,
\[W=2\times 2+\dfrac{\left( 2+3 \right)\times 1}{2}=6.5J\]
It has been already mentioned that the work done will be equivalent to the change in kinetic energy of the particle. Therefore we can write that,
\[W=\Delta KE=6.5J\]
Where \[\Delta KE\] be the change in kinetic energy. The change in kinetic energy is the difference between the final kinetic energy and the initial kinetic energy. That is,
\[\Delta KE=K{{E}_{f}}-K{{E}_{i}}\]
As the initial kinetic energy will be zero,
\[K{{E}_{i}}=0\]
Therefore, the final kinetic energy is given as,
\[K{{E}_{f}}=6.5J\]
Therefore, the final kinetic energy is found to be as option A.

Note:
Transferring the energy will be in the form of force. This amount of energy which has been transferred by the force for the motion of a body is referred to as work or work done. Hence the relation between work and energy will be direct.