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A particle executes the motion described...

A particle executes the motion described by `x (t) =x_(0) (w-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 (t0 , a (t). Show that ` x (t) and a (t0 increase with time and ` v (t)` decreases with time.

Text Solution

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` x (t) =x_(0) [1 -e^(gamma t)]
` v(t) = (dx (t))/(dt) = x_(0) gammae^(gamma t)`
`a (t) = (d v (t) )/(dt) =- x_0 gamma^2 e^(-gamma t)`
(a) When `t=0, x(t) =x_0 (1-e^(-0)) =0`
Thus, particle starts with velocity ` gamma x_0` from origin at time ` t=0`.
(b) ` x (t) is ,aximum, when ` t= prop`. Then ` x (t) = x+0`.
` x (t) is minimum, when ` t=0` a(t) =0` ` v (t) is maximum, when ` t= 0 , v (0) = x_0 gamma`
`v (t) si minimum, when ` t = prop , ` v(prop) =0` a (t0 si maximum, when ` t = prop , a (prop) =0`
` a (t) si mimimum, when ` t=0` a (0) =- x_0 gamma^2`
From above we note that ` x (t) and a (t)` inreases with time but ` v (t)` decreases with time.
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