# Question c5b76

Jun 9, 2015

$\overline{v} = \frac{2 {v}_{0} \cdot \left({v}_{1} + {v}_{2}\right)}{2 {v}_{0} + {v}_{1} + {v}_{2}}$

#### Explanation:

Average velocity is defined as total distance travelled divided by the total time needed to travel said distance.

You know that the body travels half of the total distance with a velocity of ${v}_{0}$. Let's say that it takes the body a total time of ${t}_{i}$ to travel half of the total distance.

The other half of the total distance is travelled with ${v}_{1}$ for half of the time, ${t}_{1}$, and with ${v}_{2}$ for the other half of the time, ${t}_{2}$. Let's say that the total time it took the body to travel the second half of the total ditance is ${t}_{i i}$.

You know that

$\text{distance" = "velocity" * "time}$

You can use the fact that the two halves of the total distance are equal to write

${\underbrace{{v}_{0} \cdot {t}_{i}}}_{\textcolor{b l u e}{\text{first half")) = underbrace(v_1 * t_1 + v_2 * t_2)_(color(blue)("second half}}}$

But since t_1 = t_2 = "t_(ii)/2#, you can write

${v}_{0} \cdot {t}_{i} = {v}_{1} \cdot {t}_{i i} / 2 + {v}_{2} \cdot {t}_{i i} / 2$

${v}_{0} \cdot {t}_{i} = {t}_{i i} / 2 \cdot \left({v}_{1} + {v}_{2}\right)$ $\text{ } \textcolor{b l u e}{\left(1\right)}$

This means that the average velocity can be written as

$\overline{v} = {\overbrace{{v}_{0} \cdot {t}_{i} + {t}_{i i} / 2 \left({v}_{1} + {v}_{2}\right)}}^{\textcolor{g r e e n}{\text{total distance"))/underbrace((t_i + t_(ii)))_(color(red)("total time}}}$

Use equation $\textcolor{b l u e}{\left(1\right)}$ to replace ${v}_{0} \cdot {t}_{i}$

$\overline{v} = \frac{{t}_{i i} / 2 \left({v}_{1} + {v}_{2}\right) + {t}_{i i} / 2 \left({v}_{1} + {v}_{2}\right)}{{t}_{i} + {t}_{i i}}$

$\overline{v} = \frac{{t}_{i i} \left({v}_{1} + {v}_{2}\right)}{{t}_{i} + {t}_{i i}}$

Use equation $\textcolor{b l u e}{\left(1\right)}$ again to express ${t}_{i}$ in terms of the other parameters

${t}_{i} = {t}_{i i} / \left(2 {v}_{0}\right) \cdot \left({v}_{1} + {v}_{2}\right)$ $\text{ } \textcolor{b l u e}{\left(2\right)}$

Use equation $\textcolor{b l u e}{\left(2\right)}$ into the main equation to get

$\overline{v} = \frac{{t}_{I i} \left({v}_{1} + {v}_{2}\right)}{{t}_{i i} / \left(2 {v}_{0}\right) \cdot \left({v}_{1} + {v}_{2}\right) + {t}_{i i}}$

$\overline{v} = \frac{\cancel{{t}_{i i}} \left({v}_{1} + {v}_{2}\right)}{\cancel{{t}_{i i}} \left(\frac{{v}_{1} + {v}_{2}}{2 {v}_{0}} + 1\right)}$

Thus,

$\overline{v} = \frac{{v}_{1} + {v}_{2}}{\frac{2 {v}_{0} + {v}_{1} + {v}_{2}}{2 {v}_{0}}} = \textcolor{g r e e n}{\frac{2 {v}_{0} \left({v}_{1} + {v}_{2}\right)}{2 {v}_{0} + {v}_{1} + {v}_{2}}}$