Question

A vehicle travels half of the distance with speed ${V_1}$ and the other half of the distance in two equal intervals with speed ${V_2}$ and ${V_3}$. The average speed of the vehicle is
$(1)\dfrac{{2{V_1}{V_2} + 2{V_1}{V_3}}}{{{V_1} + {V_2} + {V_3}}}$
$(2)\dfrac{{2{V_1}{V_2} + 2{V_1}{V_3}}}{{2{V_1} + {V_2} + {V_3}}}$
$(3)\dfrac{{3{V_1}{V_2}{V_3}}}{{{V_1}{V_2} + {V_2}{V_3} + {V_1}{V_3}}}$
$(4)\dfrac{{2{V_1}{V_2} + {V_1}{V_3}}}{{2{V_1} + {V_2} + {V_3}}}$

Answer 1

The average speed is simply calculated by the ratio of the total distance covered by a body to the total time of the whole journey. Here in the problem, the journey is divided into parts. So, the time taken for each part has to be calculated assuming a total distance with the help of the given speeds of the parts. Note that, for the second half the intervals are equal. With this data, the distances for the parts can be found. The simplification for this problem may be quite complicated due to fractions, so carefulness is a must.

Formula used:
The average speed $V = \dfrac{{{S_1} + {S_2} + {S_3} + ........({\text{total distance)}}}}{{{t_1} + {t_2} + {t_3} + ............({\text{total time)}}}}$

Complete step by step solution:
Let us consider, the total distance be $S$
Now for the first half distance i.e. for the distance $\dfrac{S}{2}$ , the speed is given ${V_1}$
So the time is taken for the 1st half distance, ${t_1} = \dfrac{S}{{2{V_1}}}..........(1)$
For the second half, the total distance will be $\dfrac{S}{2}$
But this distance is again divided into two parts having equal times or intervals with velocities given by ${V_2}$ and ${V_3}$.
Let, the equal time is ${t_2}$ and the parts of distances are ${S_1}$ and ${S_2}$. i.e. ${S_1} + {S_2} = \dfrac{S}{2}............(2)$
So, ${S_1} = \;{V_2}{t_2}..........(3)$ and, ${S_2} = \;{V_3}{t_2}.............(4)$
By adding the eq. (3) and (4),
$\Rightarrow {S_1} + {S_2} = {t_2}({V_2} + {V_3})$
$\Rightarrow \dfrac{S}{2} = {t_2}({V_2} + {V_3})$[by eq. (2)]
$\Rightarrow {t_2} = \dfrac{S}{{2({V_2} + {V_3})}}$
So, the total time for the second half of the journey will be,
$\Rightarrow 2{t_2} = \dfrac{S}{{({V_2} + {V_3})}}..........(5)$
Now,
The average speed,
$V = \dfrac{S}{{{t_1} + {t_2} + {t_2}}}$
$\Rightarrow V = \dfrac{S}{{{t_1} + 2{t_2}}}$
$\Rightarrow V = \dfrac{S}{{\dfrac{S}{{2{V_1}}} + \dfrac{S}{{({V_2} + {V_3})}}}}$[by, eq. (1) and (5)]
$\Rightarrow V = \dfrac{1}{{\dfrac{1}{{2{V_1}}} + \dfrac{1}{{({V_2} + {V_3})}}}}$
$\Rightarrow V = \dfrac{1}{{\dfrac{{{V_2} + {V_3} + 2{V_1}}}{{2{V_1}({V_2} + {V_3})}}}}$
$\Rightarrow V = \dfrac{{2{V_1}({V_2} + {V_3})}}{{{V_2} + {V_3} + 2{V_1}}}$
$\Rightarrow V = \dfrac{{2{V_1}{V_2} + 2{V_1}{V_3}}}{{2{V_1} + {V_2} + {V_3}}}$
So, the right answer is in Option: (4).

Note: Average speed is determined by dividing the sum of the distances that something has covered by the total quantity of intervals that took to cover those distances.
Speed is how fast a body is traveling at a certain moment. The Average speed calculates the average rate of speed over the extent of a trip. It is usually applied to transports like cars, trains, and airplanes. It is generally measured in miles per hour (mph) or kilometers per hour (kmph).
Average speed is applicable for all types of fields, including physics, astronomy, and transportation.