The displacement x of a particle varies with time according to the relation `x=(a)/(b)(1-e^(-bt))`. Then select the false alternative.

Options

A. At `t = (1)/(b)` the displacement of the particle is nearly `((2)/(3)) ((a)/(b))`
B. The velocity and acceleration of the particle at `t = 0` are a and - ab respectively
C. The particle cannot reach a point at a distance x' from its starting position such that `x' gt a//b`
D. The particle will come back to its starting point as `t rarr prop`

Answer

A, B, C

Explanation

`v = ae^(-bt)`
`alpha = -abe^(-bt)`
`t = 1//b`
`x = (a)/(b) (1 - e^(-1))`
`~~ (a)/(b) (1 - (1)/(3)) ( :. e^(-1) ~= (1)/(3))`
(A) correct `~~ (2)/(3) (a)/(b)`
At `t = 0 " " v = a`
`alpha = - ab` (B)
Also disp. Is max. when `t rarr prop rArr x_(max) = (a)/(b)`
`(1 - e^(prop)) = (a)/(b)`