A particle executes the motion described by `x (t)=x_(0) (1-e^(-gamma t)) , t gt =0, x_0 gt 0`.
(a) Where does the particle start and with what velocity ?
(b) Find maximum and minimum values of ` x (t) , a (t)`. Show that ` x (t) and a (t)` increase with time and ` v(t)` decreases with time.

Explanation

Given `x(t) = x_(0)(1-e^(-gammat))`
`v(t) = (dx(t))/(dt) = x_(0)gammae^(-gammat)`
`a(t) = (dv(t))/(dt) = -x_(0)gamma^(2)e^(-gammat)`
(a) When `t = 0, x(t) = x_(0)(t-e^(-0)) = x_(0)(1-1) = 0`
`x(t=0)=x_(0)gammae^(-0) = x_(0)gammae^(-0) = x_(0)gamma(1) = gammax_(0)`
(b) `x(t)` is maximum when `t = oo` , `[x(t)]_(max) = x_(0)`
`x(t)` is minimum when `t = 0` , `[x(t)]_(min) = 0`
`v(t)` is maximum when `t = 0, v(0) = x_(0)gamma`
`v(t)` is minimum when `t = oo, v(oo) = 0`
`a(t)` is maximum when `t = oo, a(oo) = 0`
`a(t) ` is minimum when `t = 0, a(0) = -x_(0)gamma^(2)`