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# A vehicle travels half the distance L with speed ${v_1}$ and the other half with speed ${v_2}$, then its average speed is :A) $\dfrac{{{v_1} + {v_2}}}{2}$B) $\dfrac{{2{v_1} + {v_2}}}{{{v_1} + {v_2}}}$C) $\dfrac{{2{v_1}{v_2}}}{{{v_1} + {v_2}}}$D) $\dfrac{L\left( {{v}_{1}}+{{v}_{2}} \right)}{{{v}_{1}}{{v}_{2}}}$

Last updated date: 15th Sep 2024
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Hint: The speed for the whole journey is referred to a term known as average speed. The average speed is the ratio of the total distance travelled and the total time taken for the distance to be travelled. This definition must be applied to calculate the average speed.

Complete step by step solution:
The speed is defined as the ratio of the distance travelled per unit time.
$s = \dfrac{d}{t}$
The term average speed gives us a sense of the rate at which the journey is covered. It represents the arithmetic mean of all the values of speed that the body travels over a distance.
If a body covers ${d_1}$ distance in time ${t_1}$ , ${d_2}$ distance in time ${t_2}$ and so on, the average speed is given by the expression,
$S = \dfrac{{{d_1} + {d_2} + \cdot \cdot \cdot }}{{{t_1} + {t_2} + \cdot \cdot }}$
Let D be the total distance travelled by the vehicle. One half of this distance travelled is equal to L.
Hence, we have –
$L = \dfrac{D}{2}$
Total distance in terms of L, is given by –
$D = 2L$
The total distance travelled by the car in each trip –$L$
If the forward journey takes place with the speed of ${v_1}$, the time taken for the journey is given by –
${t_1} = \dfrac{L}{{{v_1}}}$
Similarly, if the other half of the journey takes place with the speed of ${v_2}$, the time taken for the journey is given by –
${t_2} = \dfrac{L}{{{v_2}}}$
Calculating the average speed for the journey, we have –
$S = \dfrac{D}{{{t_1} + {t_2}}}$
Substituting the values, we have –
$S = \dfrac{{2L}}{{\dfrac{L}{{{v_1}}} + \dfrac{L}{{{v_2}}}}}$
$\Rightarrow S = \dfrac{{2L}}{{\dfrac{{L{v_1} + L{v_2}}}{{{v_1}{v_2}}}}}$
$\Rightarrow S = \dfrac{{2L}}{{\dfrac{{L{v_1} + L{v_2}}}{{{v_1}{v_2}}}}}$
$\Rightarrow S = \dfrac{{2L \times {v_1}{v_2}}}{{L\left( {{v_1} + {v_2}} \right)}}$
$\therefore S = \dfrac{{2{v_1}{v_2}}}{{{v_1} + {v_2}}}$

Hence, the correct option is Option C.

Note: We can see from the solution of this problem that the average speed does not include the value of L. Thus, we can say that the total average speed is mainly dependent on the velocities irrespective of the distance covered in these speeds.