Question

If the velocity of a moving particle is $v \propto {x^n}$ where x is the displacement, then:
A. when x = 0, the velocity and acceleration are zero.
B. $n > \dfrac{1}{2}$
C. $n < \dfrac{1}{2}$
D. (a) and (c) are correct

Answer 1

Hint: When a function indicates the position of something as a function of time, then the first derivative gives its velocity, and the second derivative gives the acceleration. So, when we differentiate position to get velocity, and then differentiate velocity to find the acceleration.

Formula used: $\dfrac{{dv}}{{dt}} = a$

Complete step by step solution:
Velocity gives us information about the rate of change of the position of the body, meaning how fast the position of the body is changing per unit time. In physics, the velocity is defined as the displacement divided by time where displacement is defined as the difference between the final and initial positions of the body. Further, when an object travels the same distance every second, then the object is said to be moving with constant velocity. Which describes that the magnitude of the speed or velocity and the direction of the velocity both remain constant.
 Now, from the question,
$v \propto {x^n}$
$ \Rightarrow v = k{x^n}$ (when x = 0)
$ \Rightarrow v = 0$
Now,
$ \Rightarrow v = k{x^n}$
$\eqalign{
  & \Rightarrow \dfrac{{dv}}{{dt}} = kn{x^{n - 1}}\dfrac{{dx}}{{dt}} \cr
  & \Rightarrow \dfrac{{dv}}{{dt}} = kn.0 \cr
  & \Rightarrow a = 0 \cr} $
Thus, when x = 0, then a = 0.
Now,
${v^2} - {u^2} = 2as$
$ \Rightarrow {v^2} = 2as + {u^2}$ (as initial = 0)
$\eqalign{
  & \Rightarrow {v^2} = {u^2} + 2as \cr
  & \Rightarrow {v^2} = 2as \cr} $
$\eqalign{
  & \Rightarrow v \propto \sqrt x \cr
  & \Rightarrow v = \sqrt {{u^2} + 2as} \cr
  & \Rightarrow v = {\left( {{u^2} + 2as} \right)^{1/2}} \cr
  & \therefore n < \dfrac{1}{2} \cr} $
Thus, when the velocity of a moving particle is $v \propto {x^n}$ where ‘x’ is the displacement, then when x = 0, the velocity and acceleration are zero and $n < \dfrac{1}{2}$ . Therefore, from the given options both option (a) and (c) are correct.

Hence, option (D) is the correct answer.

Note:
Acceleration of a moving object is set by identifying two states of the moving object. Which gives us the initial velocity and time, and thus the final velocity and time. Once the two states are determined, then the formula of acceleration can be applied. i.e. $a = \dfrac{{v - u}}{t}$