A particle executes the motion described by $x(t) = x_0 (1 - e^{-\gamma t} )$ ; જ્યાં $t\, \geqslant \,0\,,\,{x_0}\, > \,0$.
$(a)$ 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 $a(t)$ increase with time and $v(t)$ decreases with time.
A particle executes the motion described by $x(t) = x_0 (1 - e^{-\gamma t} )$ ; જ્યાં $t\, \geqslant \,0\,,\,{x_0}\, > \,0$.
$(a)$ 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 $a(t)$ increase with time and $v(t)$ decreases with time.
Given, $\quad x(t)=x_{0}\left(1-e^{-\gamma t}\right)$
$v(t)=\frac{d x(t)}{d t}=x_{0} \gamma e^{-\gamma t}$
$a(t)=\frac{d v(t)}{d t}=-x_{0} \gamma^{2} e^{-\gamma t}$
$(a)$ When $t=0 ; x(t)=x_{0}\left(1-e^{-0}\right)=x_{0}(1-1)=0$
$v(t)=x_{0} \gamma e^{-0}=x_{0} \gamma(1)=\gamma x_{0}$
$(b)$ $x(t)$ is maximum when $\mathrm{t}=\infty$
$[x(t)]_{\max }=x_{0}(1-0)=x_{0}$
$x(t)$ is minimum when $\mathrm{t}=0$
$[x(t)]_{\min }=0$
$v(t)$ is maximum when $t=0 ; v(0)=x_{0} \gamma$
$v(t)$ is minimum when $t=\infty ; v(\infty)=0$
$a(t)$ is maximum when $t=\infty ; a(\infty)=0$
$a(t)$ is minimum when $\mathrm{t}=0 ; a(0)=-x_{0} \gamma^{2}$
Similar Questions
Refer to the graph in figure. Match the following
Graph
Characteristics
$(A)$
$(i)$ has $v > 0$ and $a < 0$ throughout
$(B)$
$(ii)$ has $x > 0,$ throughout and has a point with $v = 0$ and a point with $a = 0$
$(C)$
$(iii)$ has a point with zero displacement for $t > 0$
$(D)$
$(iv)$ has $v < 0$ and $a > 0$
Refer to the graph in figure. Match the following
| Graph | Characteristics |
| $(A)$ | $(i)$ has $v > 0$ and $a < 0$ throughout |
| $(B)$ | $(ii)$ has $x > 0,$ throughout and has a point with $v = 0$ and a point with $a = 0$ |
| $(C)$ | $(iii)$ has a point with zero displacement for $t > 0$ |
| $(D)$ | $(iv)$ has $v < 0$ and $a > 0$ |
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A train starting from rest travels the first part of its journey with constant acceleration $a$ , second part with constant velocity $v$ and third part with constant retardation $a$ , being brought to rest. The average speed for the whole journey is $\frac{{7v}}{8}$. The train travels with constant velocity for $...$ of the total time
Figure shows the position of a particle moving on the $x$-axis as a function of time
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Given figure shows the $x-t$ plot of one-dimensional motion of a particle. Is it correct to say from the graph that the particle moves in a straight line for $t < 0$ and on a parabolic path for $t >0$? If not, suggest a suitable physical context for this graph.
A particle initially at rest moves along the $x$-axis. Its acceleration varies with time as $a=4\,t$. If it starts from the origin, the distance covered by it in $3\,s$ is $...........\,m$
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