Question

The kinetic energy K of a particle moving in a straight line depends upon the distance s as
$K = a{s^2}$. The force acting on the particle is
A. $2as$
B. $2mas$
C. $2a$
D. $\sqrt {a{s^2}} $

Answer 1

Hint: Kinetic energy is the energy stored in the body by virtue of its motion.
But what is energy?
Energy is the tendency to do work. Which means that the amount of work done on the body is stored in that body in potential form, and that constitutes the energy of the body.
Let us assume, a force F produces an infinitesimally small displacement, $ds$. The work done will be also infinitesimally small.
$dW = F.ds$
Since the work done is equal to the energy stored in the body, if loss of energy is neglected,
So,
$dW = dE = F.ds$, Work done is the dot product of force and displacement.

$
  dE = Fds\cos \theta ,{\text{ where }}\theta {\text{ is the angle between the direction of force and direction of displacement}}{\text{.}} \\
  \because {\text{the body is moving on the straight line, so the angle between the direction of force and direction of displacement is 0 degrees}}{\text{.}} \\
   \Rightarrow dE = F \times ds \times \cos 0 \\
   \Rightarrow dE = Fds \\
   \Rightarrow \dfrac{{dE}}{{ds}} = F \\
$
Which means that the derivative of Energy function with respect to the displacement will give the force acting on the body.

Complete step by step solution:
We have the following information provided in the question:
1. The kinetic energy of the body is, $K = a{s^2}$
2. The body is moving on a straight line.
The Force acting on the body can be calculated as discussed above-
The derivative of energy with respect to displacement will give us the force acting on the body.
Saying so,
$
  F = \dfrac{{dK}}{{ds}} \\
   \Rightarrow F = \dfrac{{d\left( {a{s^2}} \right)}}{{ds}}{\text{ ,using the differentiation rule, }} \\
  {\text{ }}\dfrac{{d\left( {{x^n}} \right)}}{{dx}} = n{x^{n - 1}} \\
   \Rightarrow F = a\dfrac{{d\left( {{s^2}} \right)}}{{ds}},{\text{ , since }}a{\text{ is a constant}}{\text{.}} \\
   \Rightarrow F = a(2s) \\
   \Rightarrow F = 2as \\
$

$\therefore {\text{ option A is the right option}}$

Note: Force and displacements are the vector quantities, whereas the work or energy are the scalar quantities.
Work done is the dot product of Force and displacement.
$
  \therefore W = F.s \\
   \Rightarrow W = F \times s \times \cos \theta \\
$
Since in the question it was given that the body moves in a straight line, so the angle between the force and displacement are along one line, which means the angle between the force and the displacement is 0 degrees.