A particle moves with an initial velocity `v_(0)` and retardation alphav, where alpha is a constant and v is the velocity at any time t. If the particle is at the origin at `t=0`, find
(a) (i) velocity as a function of time t.
(ii) After how much time the particle stops?
(iii) After how much time velocity decreases by `50%`?
(b) (i) Velocity as a function of displacement s.
(ii) The maximum distance covered by the particle.
(c ) Displacement as a function of time t.

Explanation

(a) (i) Retardation
`a=alphav`
`(dv)/(dt)=-alphav`
`overset(v)underset(v_(0))(int)(dv)/(v)=-alphaoverset(t)underset(0)intdt`
`|log_e v|_(v_(0))^(v) v = - alpha|t|_(0)^(t)`
`log_(e)v -log_(e)v_(0)=-alphat`
`log_(e)((v)/(v_(0)))=-alphat`
`(v)/(v_(0))=e^(-alphat)`
`v=v_(0)e^(-alphat)`
(ii) When the particle stops, i.e. `v=0, e^(-alphat)=0rArre^(alphat)=infty`
`rArr t-infty`, the particle will stops after a long time `(t=infty)`
(iii) 50% of initial velocuty, i.e. `v=(v_(0))/(2)`
`v=v_(0)e^(-alphat)`
`(v_(0))/(2)=v_(0)e^(-alphat)`
`e^(-alphat)=(1)/(2)rArr e^(alphat)=2`
Taking log on the base e, we get
`log_(e)e^(alphat)=log_(e)2`
`alpharlog_(e)2=log_(e)2`
`alphat=log_(e)2`
`alphat=log_(e)2`
`t=(log_(e)2)/(alpha)=(In 2)/(alpha)" "(log_(e)-=1n)`
(b) (i) Retardation `a=alphav`
`v(dv)/(ds)= -alphav" "[("Velocity is asked in"),("terms of displacement")]`
`overset(v)underset(v_(0))int dv= -alphaoverset(S)underset(0)int ds`
`|v|_(v_0)^(v)=-alpha|s|_(0)^(S)`
`v-v_(0)=-alpha(s-0)`
`v=v_(0)-alphas`
(ii) When the particle stops, v=0
`0=v_(0)-alphasrArrs=(v_(0))/(alpha)`
The particle will stops at distance `(v_(0))/(alpha)`
(c) `v=v_(0)e^(-alphat)`
`(ds)/(dt)=v_(0)e^(-alphat)`
`overset(s)underset(0)intds=v_(0)overset(t)underset(0)e^(-alphat)dt`
`|s|_(0)^(S)=v_(0)(|e^(-alphat)|_(0)^(t))/(-alpha)`
`s-0=-(v_(0))/(alpha)(e^(-alphat)-e^(0)`
`=-(v_(0))/(alpha)(e^(alphat)-1)`
`s=(v_(0))/(alpha)(1-e^(-alphat))`