The displacement (x) of particle depends on time (t) as
`x = alpha t^(2) - beta t^(3)`.

Options

A. The particle will return to its starting point after time `alpha//beta`
B. The particle will come to rest after time `2alpha//3beta`
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 = alpha//3beta`

Answer

A, B, C, D

Explanation

`x= alpha t^(2) - beta t^(3)`
for `x = 0, alpha t^(2) - beta t^(3) = 0`
`:. t = (alpha)/(beta)`
`(dx)/(dt) = 2 alpha t - 3 beta t^(2)`
Particle at rest `v = 0`
`2 alpha t - 3 beta t^(2) = 0`
`t = (2)/(3) (alpha)/(beta)`
`(d^(2)x)/(dt^(2)) = 2 alpha - 6 beta t`
at `t = 0 " " a = 2 alpha, " at " t = 0, v = 0`
for no net force, `(a = 0)`
`2 alpha - 6 beta t = 0, t = (alpha)/(3 beta)`