A particle is moving along the x-axis whose instantaneous speed is given by ${v^2} = 108 - 9{x^2}$. The acceleration of the particle is:
(A) $- \dfrac{{9x}}{2}m{s^{ - 2}}$
(B) $- 18xm{s^{ - 2}}$
(C) $- \dfrac{{9x}}{2}m{s^{ - 2}}$
(D) None of these

Hint We know that when the speed of an object is constantly changing, the instantaneous speed is the speed of an object at a particular moment or we can say instant in time. Based on this concept we have to solve this question, by derivation of the given equation.

Complete step by step answer
We know that it is given:
${v^2} = 108 - 9{x^2}$
So, we can write this as:
$\dfrac{{dv}}{{dx}}\dfrac{{dx}}{{dt}} = v\dfrac{{dv}}{{dx}}$
As we know a = $\dfrac{{dv}}{{dx}}\dfrac{{dx}}{{dt}} = v\dfrac{{dv}}{{dx}}$
So, the value of a = $- 18xm{s^{ - 2}}$.

Hence, option B is correct.

Note For solving such problems we know that the rate of change of distance of an object with the respect to time, is an idea that is obtained from the concept of instantaneous speed. The unit of this physical quantity is given as distance divided by time.