Question

if the position of a particle of any instant t is given by x = t^3find the acceleration of the particle at t = 2 seconds​

Answer 1

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

Position equation of particle has been provided.

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

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

We have to find acceleration of particle at t = 2s.

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

✪ 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 :

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

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

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

Putting t = 2, we get

$:\implies\tt\:a=6(2)$

$:\implies\boxed{\bf{\red{a=12\:ms^{-2}}}}$

Answer 2

Given ,

The position of a particle of any instant t is given by x = t³

We know that , the instanteous velocity is given by

$\boxed{ \sf{Velocity \: (v) = \frac{dx}{dt} }}$

Thus ,

v = d(t³)/dt

v = 3t²

Now , the instanteous acceleration is given by

$\boxed{ \sf{Acceleration \: (a) = \frac{dv}{dt} }}$

Thus ,

a = d(3t²)/dt

a = 6t

Put t = 2 sec , we get

a = 6 × 2

a = 12 m/s²

∴The acceleration of the particle at 2 sec is 12 m/s²