Which of the following option correctly describes the variation of the speed ` ` and acceleration `'a'` of a point mass falling vertically in a viscous medium that applies a force `F=-kv`, where `'k'` is constant, on the body?
(Graphs are schematic and drawn to scale)

Options

A.
B.
C.
D.

Answer

b

Explanation

Let the mass of the body be m,
Then the acceleration of the body due to viscous force can be expressed as:
`a_(1)=(F)/(m)=-(kv)/(m)`
As we know that the acceleration and the velocity are related as:
`a=(dv)/(dt)`
Or,
`(dv)/(dt)=g-(kv)/(m)`
`(dv)/(g-(kv)/(m))=dt`
Integrating both sides
In `((g-(kv)/(m))/(g))=-(k)/(m)t`
Let `(k)/(m)=a`
In `((g-av)/(g))=-at`
`((g-av)/(g))=e^(-at)`
`rArr g-av=ge^(-at)`
`rArr v=(g)/(a)(1-e^(-at))`
`a=(dv)/(dt)`
`rArr a=(g)/(a)(ae^(-at))`
`rArr a=ge^(-at)`
From equation i and ii we can conclude that the appropriate graph is as follows: