A particle moves with an initial `v_(0)` and retardation alphav, where v is its velocity at any time t.
(i) The particle will cover a total distance `v_(0)/alpha`.
(ii) The particle will come to rest after time `1/alpha`.
(iii) The particle will continue to move for a very long time.
(iv) The velocity of the particle will become `v_(0)/2` after time `(1n2)/alpha`

Options

A. (i), (ii)
B. (ii), (iii)
C. (i), (ii), (iv)
D. All

Answer

C

Explanation

`(dv)/(dt)=-alpha v`
`int_(v_(0))^(v)(dv)/v=-alphaint_(0)^(t) dt`
`| log_(e)v|_(0)^(v)=-alpha|t|_(0)^(t)`
`log_(e)v-log_(e)v_(0)=-alpha t`
`log_(e)(v//v_(0))=-alpha t`
`v//v_(0)=e^(-alpha t)`
Velocity in terms of t
`v=v_(0)e^(-alphat) ...(i)`
`(ds)/(dt)=v_(0)e^(-alphat)`
`int_(0)^(s) ds=v_(0)int_(0)^(t) e^(-alphat)dt`
`s=v_(0)|e^(-alphat)|_(0)^(t)/-alpha=v_(0)/-alpha(e^(-alphat)-e^(0))`
`s=v_(0)/alpha(1-e^(-alphat)) ...(ii)`
`v=0impliese^(-alphat)=0impliest=oo`, (iii) is O.K.
At `t=oo, s=v_(0)/alpha,` (i) is O.K.
`v=v_(0)e^(-alphat)impliesv_(0)/2=v_(0) e^(-alphat)`
`e^(alphat)=2impliesalphat log_(e)e=log_(e)2`
`t=(loge^(2))/alpha=(In 2)/alpha`, (iv) is O.K.