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PART - 1 CH - 2 Motion in a Straight Line — Physics STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 SciencePhysicsPART - 1 CH - 2 Motion in a Straight Line5 Marks
Question
We have carefully distinguished between average speed and magnitude of average velocity. No such distinction is necessary when we consider instantaneous speed and magnitude of velocity. The instantaneous speed is always equal to the magnitude of instantaneous velocity. Why?
✓

Answer

The instantaneous velocity of an object is equal to the velocity of the object at that instant is called the limiting instantaneous speed of the mean velocity. Thus, when the time interval is minute, the magnitude of displacement is effectively equal to the distance covered by the object in the same minute time interval. Hence in this case instantaneous velocity and instantaneous speed are equal. This can be understood by the following :
Image
The instantaneous velocity of a moving object will be equal to its average velocity if the interval $\Delta t$ between its two times $( t$ and $t +\Delta t$ ) is infinitely small.
Mean velocity over time interval $\Delta t$
$
=\frac{\Delta x}{\Delta t}
$
Instantaneous velocity at instant t
$
\begin{array}{r}
\qquad \lim _{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t}=\frac{d x}{d t} \quad\quad...(1)\\
\text { Mean speed }=\frac{\text { Total distance }}{\text { Total time }}=\frac{\text { Arc PQ }}{\Delta t}
\end{array}
$
$
\begin{aligned}
\text { Instantaneous speed } & =\lim _{\Delta t \rightarrow 0} \text { mean speed } \\
& =\lim _{\Delta t \rightarrow 0} \frac{Arc PQ}{\Delta t}=\lim _{\Delta t \rightarrow 0} \frac{PQ}{\Delta t} \\
& =\lim _{\Delta t \rightarrow 0} \frac{PQ}{\Delta t}\quad\quad...(2)
\end{aligned}
$
Here $PQ = PQ$ is length of line.
From $\triangle PQR ,( PQ )^2=( PR )^2+( QR )^2$
like $\Delta t \rightarrow 0, PQ \rightarrow QR$ or $PQ \rightarrow \Delta x$
From equation (2) instantaneous speed $=\lim _{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t}=\frac{d x}{d t}$
i.e., magnitude of instantaneous velocity.

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