Question

The displacement of particle is given by
$ x = a_0 + \frac{a_1t}{2}\frac{a_2t^2}{3}$
What is its acceleration ?

• A. $ \frac{2\,a_{2}}{3} $
• B. $ -\frac{2\,a_{2}}{3} $
• C. $ a_2 $
• D. zero

Answer 1

Answer: (B)

Acceleration is the rate of change of velocity and velocity is rate of change of displacement.
The displacement equation is given by
$ x = a_0 +\frac{ a_1 t}{2} - \frac{ a_2t^2}{3}$
Velocity = rate of change of displacement.
i.e., $ v = \frac{dx}{dt}$
$= \frac{d}{dt} \left(a_{0} +\frac{a_{1}t}{2} - \frac{a_{2}t^{2}}{3}\right) $
$ = 0 + \frac{a_{1}}{2} - \frac{2a_{2}t}{3} $
$ = \frac{a_{1}}{2} - \frac{2a_{2}t}{3} $
Acceleration = rate of change of velocity
i.e., $ a = \frac{dv}{dt}$
$ = \frac{d}{dt} ( \frac{a_1}{2} - \frac{2a_2}{3}t)$
$ = 0 - \frac{2a_2}{3} = - \frac{2a_2}{3}$