# An object moves with velocity of 20km/hr in the first 1/3rd of total distance and moves with 60km/hr in the second 1/3rd of total distance. What is average velocity up to 2/3 rd of total distance?

Jun 14, 2015

A car cover 1/3 distance with speed 20km/hr and 2/3 with 60km/hr. average speed is 30km/hr.

#### Explanation:

Assume total length=$L k i l o m e t e r s$
Average velocity=Total distance up to$\frac{2}{3}$part /Time up to $\frac{2}{3}$part
${V}_{a v g} = \frac{{d}_{1} + {d}_{2}}{{t}_{1} + {t}_{2}}$ $\text{ } \textcolor{b l u e}{\left(1\right)}$

where,
${v}_{1}$=velocity up to $\frac{1}{3}$rd part =20 km/hr,

${t}_{1}$=time taken for car up to ${\left(\frac{1}{3}\right)}^{r d}$ part,

${d}_{1}$=distance up to ${\left(\frac{1}{3}\right)}^{r d}$ part =$\frac{L}{3}$kilometers

${v}_{2}$=velocity from ${\left(\frac{1}{3}\right)}^{r d}$part to ${\left(\frac{2}{3}\right)}^{r d}$part=60km/hr,

${t}_{2}$=time taken from ${\left(\frac{1}{3}\right)}^{r d}$part to ${\left(\frac{2}{3}\right)}^{r d}$part,

${d}_{2}$=distance from ${\left(\frac{1}{3}\right)}^{r d}$part to ${\left(\frac{2}{3}\right)}^{r d}$part= $\frac{L}{3}$km.

For first $\frac{1}{3}$rd part:
given that,

${v}_{1} = 20 \frac{k m}{h r}$

${d}_{1} / {t}_{1} = 20$km/hr [since from velocity definition]

$\frac{L}{3} / {t}_{1} = 20$km/hr [since${d}_{1} = \frac{L}{3}$]

$\frac{L}{{t}_{1}} = 60$km/hr $\text{ } \textcolor{b l u e}{\left(2\right)}$

from ${\left(\frac{1}{3}\right)}^{r d}$part to ${\left(\frac{2}{3}\right)}^{r d}$part:

given that,

${v}_{2} = 60$km/hr,

${d}_{2} / {t}_{2}$=60km/hr,

$\frac{L}{3} / {t}_{2}$=60km/hr [since ${d}_{2} = \frac{L}{3} k m$]

$\frac{L}{3 {t}_{2}}$=60km/hr $\text{ } \textcolor{b l u e}{\left(3\right)}$

Divide equation(2) with equation(1),

We get ${t}_{1} = 3 {t}_{2}$ $\text{ } \textcolor{b l u e}{\left(4\right)}$

From equation(1),

${V}_{a v g} = \frac{{d}_{1} + {d}_{2}}{{t}_{1} + {t}_{2}}$

${V}_{a v g} = \frac{\left(\frac{L}{3}\right) + \left(\frac{L}{3}\right)}{3 \left({t}_{2}\right) + {t}_{2}}$[since from equation(4)]

${V}_{a v g} = \frac{1}{2} \cdot \left(\frac{L}{3 {t}_{2}}\right)$

${V}_{a v g} = \frac{1}{2} \cdot \left(60 \frac{k m}{h r}\right)$[since from equation (3)]

${V}_{a v g} = 30 \frac{k m}{h r}$

A car cover 1/3 distance with speed 20km/hr and 2/3 with 60km/hr. average speed is $\text{ } \textcolor{b l u e}{30}$ km/hr.