Question

A person travels along a straight line for the first half time with velocity ${{v}_{1}}$ and the second half time with velocity ${{v}_{2}}$. The mean velocity $\overline{v}$ is given by:
a)$\overline{v}=\dfrac{{{v}_{1}}+{{v}_{2}}}{2}$
b)$\dfrac{2}{v}=\dfrac{1}{{{v}_{1}}}+\dfrac{1}{{{v}_{2}}}$
c)$\overline{v}=\sqrt{{{v}_{1}}{{v}_{2}}}$
d)$\overline{v}=\sqrt{\dfrac{{{v}_{2}}}{{{v}_{1}}}}$

Answer 1

Hint: Write down the mean velocity formula and try to frame all the values in terms of known velocity and time. Also, the mean velocity is equal to the total distance travelled by the body divided by the total time taken by the body to reach the final point. Don’t get confused between average velocity and mean velocity.

Formula used:
$\begin{align}
  & \overline{v}=\dfrac{2s}{t} \\
 & {{v}_{1}}=\dfrac{s}{{{t}_{1}}} \\
 & {{v}_{2}}=\dfrac{s}{{{t}_{2}}} \\
\end{align}$

Complete step-by-step answer:
Let us assume the total distance travelled by the boy as ${{s}_{1}}+{{s}_{2}}$. The time taken to travel in two cases is $t$. Now, the speed of the boy in two cases will be ${{v}_{1}},{{v}_{2}}$.
Let us now calculate the distance travelled for first half time
$\begin{align}
  & {{t}_{{}}}=\dfrac{{{s}_{1}}}{{{v}_{1}}} \\
 & {{s}_{1}}={{v}_{1}}t \\
\end{align}$
The distance travelled for the second half time will be,
$\begin{align}
  & t=\dfrac{{{s}_{2}}}{{{v}_{2}}} \\
 & {{s}_{2}}={{v}_{2}}t \\
\end{align}$
The total distance taken by the boy will be ${{s}_{1}}+{{s}_{2}}=({{v}_{1}}+{{v}_{2}})t$
Now, the mean velocity is given by,
$\begin{align}
  & \overline{v}=\dfrac{{{s}_{1}}+{{s}_{2}}}{2t} \\
 & \overline{v}=\dfrac{({{v}_{1}}+{{v}_{2}})t}{2t} \\
 & \overline{v}=\dfrac{{{v}_{1}}+{{v}_{2}}}{2} \\
 & \\
 & \\
\end{align}$
So, the correct answer is “Option A”.

Additional Information: The average velocity of an object is its total displacement divided by the total time. It is the rate at which the body changes its position from one place to another. Also known as Average velocity, mean velocity is a vector quantity. Velocity is the rate at which the position is changing whereas average velocity will be the change in displacement or position per time ratio. There’s a lot of difference between velocity and average velocity.

Note: As per the above conditions, the mean velocity formula has been changed to required conditions, we can also find it out using distance travelled but distance travelled should be assumed and it is not given in the question, so, the solution must be written in known terms only.