Question

A body covers one third of the distance with speed ${{V}_{1}}$ and two third of the distance with speed ${{V}_{2}}$. The average speed of the body is:
A. $\dfrac{{{V}_{1}}+{{V}_{2}}}{2{{V}_{2}}}$
B. $\dfrac{{{V}_{1}}+{{V}_{2}}}{2}$
C. $\dfrac{3{{V}_{1}}{{V}_{2}}}{2{{V}_{1}}+{{V}_{2}}}$
D. $\dfrac{3{{V}_{1}}{{V}_{2}}}{{{V}_{1}}+2{{V}_{2}}}$

Answer 1

Hint: Average speed is the ratio of total distance covered by the body to the total time taken by it to cover that distance. We can calculate the above two terms with the given data and will put the values in the final formula.

Formula used:
$\text{Average speed = }\dfrac{\text{total distance}}{\text{total time}}$

Complete step by step answer:
The average speed of a body is the total distance travelled by the body divided by the elapsed time to cover that distance. Average speed is a scalar quantity. It is defined by magnitude only. To find the value of average speed, we need to know the values of total distance covered by the body and the time taken by it.
We are given that the body covers one third of the distance with speed ${{V}_{1}}$and two third of the distance with speed ${{V}_{2}}$.
Let the total distance covered by the body=$3x$
For first third distance, the velocity of body is given as ${{V}_{1}}$
Time taken for covering first third of distance ${{t}_{1}}=\dfrac{x}{{{V}_{1}}}$
For the next two third distance, the velocity of body is given as ${{V}_{2}}$
Time taken for covering next two third of the total distance ${{t}_{2}}=\dfrac{2x}{{{V}_{2}}}$
Total time taken by the body,
$\begin{align}
  & {{t}_{1}}+{{t}_{2}}=\dfrac{x}{{{V}_{1}}}+\dfrac{2x}{{{V}_{2}}} \\
 & {{t}_{1}}+{{t}_{2}}=x\left( \dfrac{1}{{{V}_{1}}}+\dfrac{2}{{{V}_{2}}} \right) \\
\end{align}$$\text{Average speed = }\dfrac{\text{total distance}}{\text{total time}}$

Total distance = $3x$
Total time =$x\left( \dfrac{1}{{{V}_{1}}}+\dfrac{2}{{{V}_{2}}} \right)$
$\text{Average speed = }\dfrac{3x}{x\left( \dfrac{1}{{{V}_{1}}}+\dfrac{2}{{{V}_{2}}} \right)}=\dfrac{3{{V}_{1}}{{V}_{2}}}{2{{V}_{1}}+{{V}_{2}}}$
$\text{Average speed = }\dfrac{3{{V}_{1}}{{V}_{2}}}{2{{V}_{1}}+{{V}_{2}}}$
Hence, the correct option is C.

Note: While calculating the time taken by body during different intervals, make sure to take the value of distance covered in that interval only, not the whole distance covered by the body. Be careful with all the units mentioned, prefer to work in SI units.