Give examples of a one - dimensional motion where
$(a)$ The particle moving along positive $x- $ direction comes to rest periodically and moves forward.
$(b) $ The particle moving along positive $x-$ direction comes to rest periodically and moves backward.
$(a)$ The particle moving along positive $x- $ direction comes to rest periodically and moves forward.
$(b) $ The particle moving along positive $x-$ direction comes to rest periodically and moves backward.
When we are writing an equation belonging to periodic nature it includes sine or cosine function.
$(a)$ The particle will be moving along positive $x$-direction only if $x(t)=1-\sin t$
Velocity $v(t)=\frac{d x(t)}{d t}=1-\cos t$
Acceleration $a(t)=\frac{d v}{d t}=\sin t$
When $t=0 ; x(t)=0$
When $t=\pi ; x(t)=\pi>0$
When $t=0 ; x(t)=2 \pi>0$
$(b)$ Equation can be represented by
$x(t)=\sin t$
$\therefore v=\frac{d}{d t} x(t)=\cos t$
As displacement and velocity includes $\sin t$ and $\cos t$ hence these equations represent periodic.
$(a)$ The particle will be moving along positive $x$-direction only if $x(t)=1-\sin t$
Velocity $v(t)=\frac{d x(t)}{d t}=1-\cos t$
Acceleration $a(t)=\frac{d v}{d t}=\sin t$
When $t=0 ; x(t)=0$
When $t=\pi ; x(t)=\pi>0$
When $t=0 ; x(t)=2 \pi>0$
$(b)$ Equation can be represented by
$x(t)=\sin t$
$\therefore v=\frac{d}{d t} x(t)=\cos t$
As displacement and velocity includes $\sin t$ and $\cos t$ hence these equations represent periodic.
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