Question

The displacement $x$ of a particle depends on time $t$ as $x=\alpha t^2-\beta t^3.$ Then,

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

Answer 1

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

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

$x=\alpha t^2-\beta t^3$
for $x$ to be zero,
$\alpha t^2-\beta t^3=0$
$\Rightarrow \frac{t^3}{t^2}=\frac{\alpha }{\beta }\Rightarrow t=\left(\frac{\alpha }{\beta }\right)$

Differentiating $x$
$v=\frac{dx}{dt}=2\alpha t-3\beta t^2$
$v_{att=0}=0$
for $v$ to be zero,
$2\alpha t=3\beta t^2\Rightarrow t=\left(\frac{2\alpha }{3\beta }\right)$
Differentiating $v$
$a=\frac{dv}{dt}=2\alpha -6\beta t$
$a_{att=0}=2\alpha$
For $a$ to be zero
$0=2\alpha -6\beta t\Rightarrow t=\frac{\alpha }{3\beta }$