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A particle executes the motion described by $x(t) = x_{0} (1 - e^{-\gamma t})$ where $t \geq 0, x0 > 0$

a) Where does the particles start and with what velocity?

b) Find maximum and minimum values of $x(t), v(t), a(t)$ . Show that $x(t)$ and $a(t)$ increase with time and $v(t)$ decreases with time.

Answers (1)

Here, $x(t)=x_{0}[1-e^{-\gamma t}]$

So, $v(t)=\frac{dx(t)}{dt}$

$=\frac{d}{dt}[x_{0}(1-e^{-\gamma t})]$

$=+x_{0}\gamma e^{-\gamma t}\; \; \; \; \; \; ..........(i)$

&

+ $a(t)=\frac{dv}{dt}$

$=-x_{0}\gamma ^{2}e^{-\gamma t}\; \; \; \; \; \; ..........(ii)$

$(i) x(0) = x_{0} [1 - e^{0}]$

$= x_{0} (1 - 1) =0$

$v(0) = x_{0}\gamma e^{0}$

$=x_{0}\gamma$

Thus, $x=0$ is the starting point of the particle and its velocity is $v_{0}=x_{0}\gamma$

(b) $x (t)\ is,$

Maximum at $t = \infty$ since $t = \infty$ $[x(t)]_{max} = \infty$

Minimum at $t = 0$ since at $t = 0$ , $[x(t)]min = 0$

v(t) is,

maximum at $t = 0$ since $t=0, v(0) = x_{0}\gamma$

minimum at $t = \infty$ since, $t = \infty$ , $v(\infty ) = 0$

a(t) is,

maximum at $t = \infty$ since at $t = \infty$ , $a(\infty )=0$

minimum at $t = 0$ since at $t = 0$ , $a(0)=-x_{0}\gamma ^{2}$

Posted by

infoexpert22

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