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A vehicle travels half the distance $l$ with speed $v_1$ and the other half with speed $v_2$, then its average speed is
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Verified Answer
The correct answer is:
$\frac{2 v_1 v_2}{v_1+v_2}$
$\frac{2 v_1 v_2}{v_1+v_2}$
Time taken to travel first half distance, $t_1=\frac{l / 2}{v_1}=\frac{l}{2 v_1} \quad\left(\because L_1=l / 2\right)$
Time taken to travel second half distance, $t_2=\frac{l}{2 v_2} \quad\left(\because L_2=l / 2\right)$
So, total time taken to travel full distance
$$
\begin{aligned}
&=t_1+t_2 \\
&=\frac{l}{2 v_1}+\frac{l}{2 v_2}=\frac{l}{2}\left[\frac{1}{v_1}+\frac{1}{v_2}\right] \\
&\text { Total time }=\frac{l}{2}\left[\frac{v_2+v_1}{v_1 v_2}\right]
\end{aligned}
$$
So, average speed,
$$
\begin{aligned}
&v_{\mathrm{av} .}==\frac{\text { Total distance }}{\text { Total time }} \\
&\Rightarrow v_{\mathrm{av} .}=\frac{l}{\frac{l}{2}\left[\frac{1}{v_1}+\frac{1}{v_2}\right]}=\frac{2 v_1 v_2}{v_1+v_2} \\
&\therefore v_{\mathrm{av} .}=\frac{2 v_1 v_2}{v_1+v_2}
\end{aligned}
$$
Time taken to travel second half distance, $t_2=\frac{l}{2 v_2} \quad\left(\because L_2=l / 2\right)$
So, total time taken to travel full distance
$$
\begin{aligned}
&=t_1+t_2 \\
&=\frac{l}{2 v_1}+\frac{l}{2 v_2}=\frac{l}{2}\left[\frac{1}{v_1}+\frac{1}{v_2}\right] \\
&\text { Total time }=\frac{l}{2}\left[\frac{v_2+v_1}{v_1 v_2}\right]
\end{aligned}
$$
So, average speed,
$$
\begin{aligned}
&v_{\mathrm{av} .}==\frac{\text { Total distance }}{\text { Total time }} \\
&\Rightarrow v_{\mathrm{av} .}=\frac{l}{\frac{l}{2}\left[\frac{1}{v_1}+\frac{1}{v_2}\right]}=\frac{2 v_1 v_2}{v_1+v_2} \\
&\therefore v_{\mathrm{av} .}=\frac{2 v_1 v_2}{v_1+v_2}
\end{aligned}
$$
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