Let the total distance of the journey be $D$. The journey is divided into two equal halves of distance $d = D/2$. Let the average speeds over these two halves be $v_1$ and $v_2$, respectively.

The average speed for the entire journey, $v_{avg}$, is defined as the total distance divided by the total time.

$$v_{avg} = \frac{D_{total}}{T_{total}}$$

The total distance is $D_{total} = d + d = 2d$. The total time, $T_{total}$, is the sum of the times taken for each half, $t_1$ and $t_2$.

$$T_{total} = t_1 + t_2$$

The time taken for each half is calculated as $t = \frac{\text{distance}}{\text{speed}}$:

$t_1 = \frac{d}{v_1}$ $t_2 = \frac{d}{v_2}$

Substituting these into the expression for total time:

$$T_{total} = \frac{d}{v_1} + \frac{d}{v_2} = d \left( \frac{1}{v_1} + \frac{1}{v_2} \right)$$

Now, we can find the average speed for the entire journey:

$$v_{avg} = \frac{2d}{d \left( \frac{1}{v_1} + \frac{1}{v_2} \right)} = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}}$$

Simplifying this expression gives the formula for the harmonic mean of the two speeds:

$$v_{avg} = \frac{2v_1 v_2}{v_1 + v_2}$$

[Q1] Substitute the given values $v_1 = 3$ m/s and $v_2 = 5$ m/s into the derived formula:

$$v_{avg} = \frac{2(3)(5)}{3 + 5} = \frac{30}{8} = 3.75 \text{ m/s}$$