Question

A body has speed V, 2V and 3V in first $\dfrac{1}{3}$ distance S, second $\dfrac{1}{3}$ of S and third $\dfrac{1}{3}$ of S respectively. Its average speed will be:
A) V
B) 2V
$\text{C)}\dfrac{18}{11}\text{V}$
$\text{D)}\dfrac{11}{18}\text{V}$

Answer 1

To calculate average speed, when time taken is not given, first we need to calculate time taken and then calculate the total time taken which can be used to calculate the average speed of the body. In case the exact value of distance is not given, assume any variable which represents distance.

Formula used:
Average speed = $\dfrac{\text{Total Distance}}{\text{Time taken}}$

Complete answer:
In this question, total distance is not given.
So, let’s assume,
Total distance = d
Now, as we know that the formula to calculate average speed is given by:
$\text{Average speed=}\dfrac{\text{Total Distance}}{\text{Time Taken}}$ -------Equation(1)
From equation (1), $\text{Time taken=}\dfrac{\text{Total distance}}{\text{Average speed}}$
Now, say ${{\text{t}}_{1}}$= time taken to travel first $\dfrac{1}{3}$ distance
${{\text{t}}_{2}}$= time taken to travel second $\dfrac{1}{3}$ distance
${{\text{t}}_{3}}$= time taken to travel third $\dfrac{1}{3}$ distance
From the question and formula we can get the values of ${{\text{t}}_{1}}$, ${{\text{t}}_{2}}$ and ${{\text{t}}_{3}}$
${{\text{t}}_{1}}=\dfrac{\text{d}}{1\times 3}$
${{\text{t}}_{2}}=\dfrac{\text{d}}{2\times 3}$
${{\text{t}}_{3}}=\dfrac{\text{d}}{3\times 3}$

Now, as we need to find the average speed, we will use formula from equation no. (1),
$\Rightarrow \text{Average speed = }\dfrac{\text{d}}{{{\text{t}}_{1}}+{{\text{t}}_{2}}+{{\text{t}}_{3}}}$
$\Rightarrow \text{Average speed = }\dfrac{\text{d}}{\dfrac{\text{d}}{1\times 3}+\dfrac{\text{d}}{2\times 3}+\dfrac{\text{d}}{3\times 3}}$
We will take ‘d; common from denominator and numerator,
$\Rightarrow \text{Average speed = }\dfrac{1}{\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{9}}$
Further solving this fraction we will get,
$\Rightarrow \text{Average speed = }\dfrac{18}{6+3+2}$
$\Rightarrow \text{Average speed = }\dfrac{18}{11}\text{V}$
Hence, we can say that the average speed of the body will be $\dfrac{18}{11}\text{V}$.

So, the correct answer is “Option C”.

Note:
It is advisable to use a variable if the actual value is not given in the question, so as to avoid calculation errors.
As, here, in these cases we have assumed, distance as ‘d’.
The average speed is equal to the total distance divided by total time taken.