Question

The displacement $x$ of a particle depend on time $t$ as $x=\alpha t^2-\beta t^3$

• A. particle will return to its starting point after time $t=\frac{\alpha }{\beta }$.
• B. the particle will come to rest after time $t=\frac{2\alpha }{3\beta }$.
• C. the initial velocity of the particle was zero but its initial acceleration was not zero.
• D. no net force act on the particle at time $t=\frac{\alpha }{3\beta }$.

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A particle will return to its starting point after time $t=\frac{\alpha }{\beta }$.
B the particle will come to rest after time $t=\frac{2\alpha }{3\beta }$.
C the initial velocity of the particle was zero but its initial acceleration was not zero.
D no net force act on the particle at time $t=\frac{\alpha }{3\beta }$.

Given, $x=\alpha t^2-\beta t^3$
Particle will return to its starting point when,$x=\alpha t^2-\beta t^3=0$
or,
$t=\frac{\alpha }{\beta }$
Velocity$=v=\frac{dx}{dt}=2t\alpha -3\beta t^2.....\left(1\right)$
at $t=0,v=0$so the initial velocity zero.
Acceleration$=a=\frac{dv}{dt}=2\alpha -6t\beta .....\left(2\right)$
at $t=0,a=2\alpha$ so initial acceleration does not zero.
The particle will come to rest when, $2t\alpha -3\beta t^2=0$ from (1)
or,$t=\frac{2\alpha }{3\beta }$
At,$t=\frac{\alpha }{3\beta },a=2\alpha -6\beta \left(\frac{\alpha }{3\beta }\right)=0$
so net force$=ma=0$ ,thus no net force act on the particle when $t=\frac{\alpha }{3\beta }$