Question

A particle of mass 2kg travels along a straight line with velocity $v = a\sqrt x $, where x is constant. The work done by net force during the displacement of particle from $x = 0$ to $x = 4m$ is
A. ${a^2}$
B. $2{a^2}$
C. $4{a^2}$
D. $\sqrt 2 {a^2}$

Answer 1

Hint: We are given with the velocity mass and displacement. So first find the acceleration using the third equation of motion where initial velocity is zero as the initial displacement is zero. Force is the product of mass and acceleration and Work done is the product of net force and the displacement of the particle. Using this info find the work done by the particle.

Complete step by step answer:
We are given the mass of a particle is 2kg and velocity v is $v = a\sqrt x $ where x is a constant.
We have to calculate the work done by net force during the displacement of particle from $x = 0$ to $x = 4m$
As the initial displacement is 0, the initial velocity u will also be zero.
Therefore substituting the values of velocities, displacement in the third equation of motion, we get
$
  {v^2} = {u^2} + 2as \\
  u = 0,s = x = 4 - 0 = 4m,a = {a_{acc}},v = a\sqrt x \\
    \Rightarrow {\left( {a\sqrt x } \right)^2} = 0 + 2{a_{acc}}\left( 4 \right) \\
   \Rightarrow {a^2}\left( x \right) = 8{a_{acc}} \\
  \therefore {a_{acc}} = \dfrac{{{a^2} \times 4}}{8} = \dfrac{{{a^2}}}{2}m/{s^2} \\
 $
Force is mass times acceleration.
$
  F = m \times {a_{acc}} \\
  m = 2kg,{a_{acc}} = \dfrac{{{a^2}}}{2}m/{s^2} \\
   \Rightarrow F = 2 \times \dfrac{{{a^2}}}{2} = {a^2}N \\
 $
Work done by a particle is the product of net force with displacement.
Therefore, word done is
$
  W = F \times d \\
  F = {a^2}N,d = 4m \\
   \Rightarrow = {a^2} \times 4 \\
  \therefore W = 4{a^2}J \\
 $
Therefore, the work done by net force is $4{a^2}$

Hence, the correct option is Option C.

Note: The work done by a body is also equal to the change in its energy. The units of work and energy are Joules or Newton-metres. The only difference between work and energy is that work is transferring an amount of energy with the help of a force. These both are scalar quantities.