Question

If the position of a particle at any instant t is given by x = t³, find the acceleration of the particle.​

Answer 1

Given that the position of a particle at any instant t is given by x = t³ , and we have to find the acceleration of the same particle at any instant.

given,

⇒ x = t³

Differentiating both sides w.r.t t , we would get velocity at amy instant because, v = dx / dt

⇒ dx / dt = d(t³) / dt

⇒ v = 3t²

Now, The acceleration of the same particle at any instant can be calculated by differentiating the resultant equation both sides w.r.t t,

⇒ dv / dt = d(3t²) / dt

⇒ a = 6t

Hence, Acceleration of the particle at any instant is given by a = 6t

Extra Information :-

▪ Velocity is the change of displacement (position) with respect to time, v = dx / dt

▪ Acceleration is the change of velocity with respect to time.

a = dv / dt

Answer 2

Explanation:

$\large{\bf{\gray{\underline{\underline{\orange{Given:}}}}}}$

Position equation of particle has been provided.

$\dag\:\boxed{\bf{x=t^3}}$

$\large{\bf{\gray{\underline{\underline{\green{To\:Find:}}}}}}$

• the acceleration of the particle.

$\large{\bf{\gray{\underline{\underline{\pink{Solution:}}}}}}$

✪ Instantaneous velocity :

$:\implies\sf\:v=(lim\:\Delta t\to 0)\:\dfrac{\Delta x}{\Delta t}=\dfrac{dx}{dt}$

$:\implies\tt\:v=\dfrac{d(t^3)}{dt}$

$:\implies\bf\:v=3t^2$

✪ Instantaneous acceleration :

$:\implies\sf\:a=(lim\:\Delta t\to 0)\:\dfrac{\Delta v}{\Delta t}=\dfrac{dv}{dt}$

$:\implies\tt\:a=\dfrac{d(3t^2)}{dt}$

$:\implies\bf\:a=6t$