Question

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

• A. the particle will return to its starting point after time α/β
• B. the particle will come to rest after time 2 α/3β
• 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 = α/3β

Answer 1

The Correct Option is D

Explanation:
$x=\alpha t^2-\beta t^3$At $t=0,x=0$Particle will come to its initial position when $x=0$ again$\alpha t^2-\beta t^3=0\Rightarrow t=\frac{\alpha }{\beta }$Velocity of the particle$v=\frac{dx}{dt}=2at-3\beta t^2$At $t=0,v=0$So, we can say that object is at rest initially.Velocity will be zero when$2at-3\beta t^2=0\Rightarrow t=\frac{2\alpha }{3\beta }$Acceleration of the particle$a=\frac{dv}{dt}=2\alpha -6\beta t$At $t=0$$a=2\alpha$Acceleration of the particle will be zero.$a=2\alpha -6\beta t=0\Rightarrow t=\frac{\alpha }{3\beta }$When the acceleration $=0,$ force acting on it will be zero.