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The potential energy of a particle of mass 5kg moving in the x – y plane is given by U = (-7x+24y) J, x and y being in meters. If the particle starts from rest from origin then speed of particle at t = 2s is
a) $5m{s^{ - 1}}$
b) $01m{s^{ - 1}}$
c) $17.5m{s^{ - 1}}$
d) $10m{s^{ - 1}}$

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Last updated date: 16th Jun 2024
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Answer
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Hint: Newton’s second law of motion states that the force is equal to the rate of change of momentum. Force is equal to the mass times acceleration, for a constant mass. The given question is an application of Newton's second law of motion. In order to find the solution of the question write down the given physical quantities and apply the formula.
Formula Used: F = ma

Complete answer:
It is given in the question that,
Mass of the particle is, m = 5kg
Initial velocity, U = 0
Time, t = 2sec
We have to find the, Final speed of the particle, i.e. v =?
The relation between the potential energy and the conservative force is given as,
$\eqalign{
  & \vec F = \dfrac{{du}}{{dx}}\hat i - \dfrac{{du}}{{dx}}\hat j - \dfrac{{du}}{{dx}}\hat k \cr
  & \Rightarrow \vec F = \dfrac{{du}}{{dx}}\left( {7x + 24y} \right) - \dfrac{{du}}{{dx}}\left( {7x + 24y} \right) - 0 \cr
  & \Rightarrow \vec F = \left[ { - 7\dfrac{d}{{dx}}\left( x \right) - 0} \right]\hat i - \left[ {7\left( 0 \right) + \dfrac{d}{{dy}}\left( {24y} \right)} \right]\hat j \cr
  & \Rightarrow \vec F = - 7\hat i - 24\hat j \cr
  & \Rightarrow F = \sqrt {{{\left( { - 7} \right)}^2} + {{\left( { - 24} \right)}^2}} \cr
  & \therefore F = 25N \cr} $
We know that force is given by the formula,
F = ma
Where ‘F’ is the force, ‘m’ is the mass of the object or the particle and ‘a’ is the acceleration. Substituting acceleration ‘a’ as, $a = \dfrac{v}{t}$
we get, $F = m\left( {\dfrac{{v - u}}{t}} \right)$
Since the initial velocity ‘u’ of the particle is zero.
Substituting the value of force ‘F’ and mass ‘m’ of the particle we get,
$\eqalign{
  & \Rightarrow 25 = 5\left( {\dfrac{{v - 0}}{2}} \right) \cr
  & \therefore v = 10m{s^{ - 1}} \cr} $
Thus, the speed of particles at t = 2s is 10m/s.

Hence, option (d) is the correct answer.

Note:
A conservative is defined as the force for which the total work done to move a particle between two given points is independent of the path taken.
Students should remember that in case of conservative forces such as gravitational force, electrostatic force, force is taken as, F = $ - \dfrac{{dU}}{{dx}}$.