A vehicle travels half the distance L with speed V1 and the other half with speed V2, then its average speed is

To find the average speed of the vehicle, we use the fundamental definition of average speed:

$\text{Average Speed}=\frac{\text{Total Distance}}{\text{Total Time Taken}}$

Let the total distance traveled by the vehicle be $L$.
According to the problem, the vehicle travels the first half of the distance, $\frac{L}{2}$, with a speed of $V_1$, and the second half of the distance, $\frac{L}{2}$, with a speed of $V_2$.

First, let's calculate the time taken to cover the first half of the distance, denoted as $t_1$:
$t_1=\frac{\text{Distance}}{\text{Speed}}=\frac{L/2}{V_1}=\frac{L}{2V_1}$

Similarly, the time taken to cover the second half of the distance, denoted as $t_2$, is:
$t_2=\frac{L/2}{V_2}=\frac{L}{2V_2}$

The total time taken for the entire journey, $t$, is the sum of $t_1$ and $t_2$:

We can simplify this by finding a common denominator:
$t=\frac{L}{2}(\frac{1}{V_1}+\frac{1}{V_2})=\frac{L}{2}\left(\frac{V_1+V_2}{V_1{V}_2}\right)=\frac{L(V_1+V_2)}{2V_1{V}_2}$

Now, we substitute the total distance $L$ and the total time $t$ back into the average speed formula:
$\text{Average Speed}=\frac{L}{t}=\frac{L}{\frac{L(V_1+V_2)}{2V_1{V}_2}}$

The term $L$ cancels out from both the numerator and denominator:
$\text{Average Speed}=\frac{2V_1{V}_2}{V_1+V_2}$

This is the harmonic mean of the two speeds, which represents the average speed when equal distances are traveled at different speeds.