Question

A particle of mass m at rest is acted upon by a force F for a time t, its kinetic energy after an interval t is:
$\left( A \right)\dfrac{{{F^2}{t^2}}}{m}$
$\left( B \right)\dfrac{{{F^2}{t^2}}}{{2m}}$
$\left( C \right)\dfrac{{{F^2}{t^2}}}{{3m}}$
$\left( D \right)\dfrac{{Ft}}{{2m}}$

Answer 1

Here we have to use the formula of force, rectilinear motion formula and kinetic energy formula. First with the help of a rectilinear motion formula we can find the acceleration of the particle. Then putting this acceleration value in the force formula we can find the velocity of the particle and then using this velocity value in the kinetic energy formula we can find the solution to this question.

Complete step by step solution:
As per the given problem we know there is a particle of mass m at rest that is acted upon by a force F for a time t.
We need to calculate the kinetic energy of the particle after an interval of time t.
From the problem we know that initially the particle is at rest and after time t it acquires a final velocity.
First of all we know the value of final velocity hence first we have to calculate this value so as use it in the kinetic energy formula
Step 1:
Using rectilinear motion formula we will get,
$v = u + at$
Where,
Initial velocity of the particle is u
Final velocity of the particle is v
Acceleration of the particle is a
Time taken by the particles is t
We know initially the particle is at rest hence $u = 0$
Using this information in rectilinear motion we will get,
$v = at$
Rearranging the equation we will get,
$a = \dfrac{v}{t} \ldots \ldots \left( 1 \right)$
Step 2:
We know force formula,
$F = ma$
Where,
Force on the particle = F
Mass of the particle is = m
Acceleration of the particle = a
We know the value of acceleration from equation $\left( 1 \right)$ ,now putting this value in the force formula we will get,
$F = m\dfrac{v}{t}$
Rearranging the above equation we will get,
$v = \dfrac{{Ft}}{m} \ldots \ldots \left( 2 \right)$
Step 3:
We know the kinetic energy formula we will get,
$K.E = \dfrac{1}{2}m{v^2}$
Where,
Kinetic energy of the particle = K.E
Mass of the paricke = m
Velocity of the particle = v
Now putting equation $\left( 2 \right)$ in place of kinetic energy formula we will get,
$K.E = \dfrac{1}{2}m{\left( {\dfrac{{Ft}}{m}} \right)^2}$
Further solving we will get,
$K.E = \dfrac{1}{2}m{\dfrac{{{F^2}t}}{{{m^2}}}^2}$
Cancelling the common terms we will get,
$K.E = \dfrac{1}{2}{\dfrac{{{F^2}t}}{m}^2}$
Hence we get the kinetic energy of the particle at an interval of t.
Therefore the correct option is $\left( B \right)$.

Note:
Always keep in mind there are three different formulas for rectilinear motion with help of given data we can find out which we have to use so as to get the required solution. Also remember that if a body is at rest at the beginning then its initial velocity must be zero.