Question

The position of a particle travelling along x-axis is given by equation x = [(t^3)/3 - (3t^2)/2 + 2t ] metres. Here t is in seconds. Find.
A) Initial velocity of particle
B) Find acceleration of particle when at rest.​

Answer 1

A) Velocity is the derivative of Displacement.

$\bold{v = \dfrac{dx}{dt}}$

$\bold{ = \dfrac{ d( \dfrac{ {t}^{3} }{3} - \dfrac{3 {t}^{2} }{2} + 2t )}{dt}}$

$\bold{= {t}^{2} - 3t + 2}$

B) Acceleration is the derivative of velocity.

Given, v = 0

$\bold{t² - 3t + 2 = 0}$

$\bold{t² - 2t - t + 2 = 0}$

$\bold{t(t - 2) - 1(t - 2) = 0}$

$\bold{(t - 1)(t - 2) = 0}$

$\bold{t = 1 (or) 2}$

$\bold{a = \dfrac{dv}{dt}}$

$\bold{ = \dfrac{d( {t}^{2} - 3t + 2)}{dt}}$

$\bold{= 2t - 3}$

$\bold{= 2(1) - 3 = - 1 }$

$\bold{ = 2(2) - 3 = 1}$