A particle executes the motion described by `x(f) =x_(0) (1- e ^(lamda t) ) , t ge 0, x_(0) gt 0.`
(a) Where does the particle start and with what velocity ? (b) Find maximum and minimum values of x(t), v(t), a(t). Show that X(t) and alt) 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) gamma e ^(- gammat )`
`a (t) = (dv(t))/(dt) =- x _(0) gamma ^(2) e ^(- gamma t)`
(a) When `t =0, x (t) = x _(0) (1- e ^(-0)) =x_(0) (1-1) =0`
`v (t) = x _(0) gamma e ^(-0) = x _(0) gamma (1) = gamma x _(0)`
(b) `x (t)` is maximum when `t = oo`
`[x (t) ] _(max) = x _(0) (1-0) = 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 maximum 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)`
Note: We should he careful about nature of variation of the curve and maximum and minimum value will be decided accordingly.