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A person travelled the first half of the journey with a velocity of $ 20 \mathrm{~m} / \mathrm{s} $ and the rest with a velocity of $ 30 \mathrm{~m} / \mathrm{s} $. Find the average velocity of the whole journey.
Given:
Velocity in the first half of the journey, $V_1$= 20 m/s
Velocity in the rest of the journey, $V_2$ = 30 m/s
To find: Average Velocity, $V_{av}$
Solution:
Let $s$ = displacement of half of the journey
We know that-
$Average\ Velocity=\frac {Total\ displacemnt}{Total\ time}$
So, we have to find out total displacement and total time.
Now,
We know that-
$Time=\frac {Displacemnt}{Velocity}$
Therefore,
Time taken in first half of the journey = $\frac {s}{20}$
Time taken in rest of the journey = $\frac {s}{30}$
$Total\ time\ taken=Time\ taken\ in\ first\ half\ of\ the\ journey\ + Time\ taken\ in\ rest\ of\ the\ journey$
$Total\ time\ taken=\frac {s}{20}+\frac {s}{30}$
$Total\ time\ taken=\frac {3s+2s}{60}$
$Total\ time\ taken=\frac {5s}{60}$
$Total\ time\ taken=\frac {s}{12}$
Now, putting the value of total time and total displacement in the average velocity formula we get-
$Average\ Velocity=\frac {s+s}{\frac {s}{12}}$
$V_{av}=\frac {2s}{\frac {s}{12}}$
$V_{av}=\frac {2s\times {12}}{s}$
$V_{av}=\frac {24s}{s}$
$V_{av}=24m/s$
Thus, the average velocity $V_{av}$ of the whole journey will be 24m/s.