Question

A person walking in a straight line, covers $\dfrac{1}{3}$part of the distance to be travelled with a speed of ${v_1}$and remaining distance with speed ${v_2}$. So the average speed is
A) \[\dfrac{{3{v_1}{v_2}}}{{{v_1} + 2{v_2}}}\]
B) $\dfrac{{3{v_1}{v_2}}}{{2{v_1} + {v_2}}}$
C) $\dfrac{{2{v_1}{v_2}}}{{{v_1} + 2{v_2}}}$
D) $\dfrac{{2{v_1}{v_2}}}{{2{v_1} + {v_2}}}$

Answer 1

Hint:Recall that the distance travelled by an object is the length of the actual path between the initial and the final position of the object or person moving in a given period of time. It is a scalar quantity. It is measured in metres. Also speed of an object is defined as the magnitude of change in its position. It is also a scalar quantity.

Complete step by step solution:
Step I:
Let the total distance travelled be ‘S’
The person covers distance $\dfrac{S}{3}$with a speed of ${v_1}$
Let ${t_1}$is the time taken to cover this distance
The remaining distance is $\dfrac{{2S}}{3}$covered with speed of ${v_2}$
Let ${t_2}$be the time taken to cover the remaining distance.
Step II:
It is known that
$Speed = \dfrac{{Dis\tan ce}}{{Time}}$
$ \Rightarrow Time = \dfrac{{Dis\tan ce}}{{Speed}}$
Therefore, ${t_1} = \dfrac{{\dfrac{S}{3}}}{{{v_1}}}$
Or ${t_1} = \dfrac{S}{{3{v_1}}}$---(i)
Similarly, ${t_2} = \dfrac{{\dfrac{{2S}}{3}}}{{{v_2}}}$
${t_2} = \dfrac{{2S}}{{3{v_2}}}$---(ii)
Step III:
Total time taken to cover the complete distance can be known by adding (i) and (ii),
${t_1} + {t_2} = \dfrac{S}{{3{v_1}}} + \dfrac{{2S}}{{3{v_2}}}$
Step IV:
Average speed is calculated by dividing the total distance covered by the total time taken to cover the distance. Therefore, average speed can be known by using formula
$Speed = \dfrac{{TotalDis\tan ce}}{{TotalTime}}$
$Speed = \dfrac{S}{{\dfrac{S}{{3{v_1}}} + \dfrac{{2S}}{{3{v_2}}}}}$
$Speed = \dfrac{S}{{\dfrac{{S({v_2} + 2{v_1})}}{{3{v_1}{v_2}}}}}$
$Speed = \dfrac{{3{v_1}{v_2}}}{{2{v_1} + {v_2}}}$
Step V:
Therefore the average speed of the person walking in a straight line is $\dfrac{{3{v_1}{v_2}}}{{2{v_1} + {v_2}}}$

$ \Rightarrow $Option B is the right answer.

Note:It is to be noted that the term distance is not to be confused with displacement because they are different. The length of the path between two points is distance but displacement is the direct length between two points when taken along a minimum path. It is a vector quantity and is represented using an arrow. It can be positive, negative or zero. Whereas distance can have only positive values.