A body travelling along a straight line traversed one third of the total distance with a\ velocity

Question

A body travelling along a straight line traversed one third of the total distance with a\ velocity $\mathrm{v}_{1}$. The remaining part of the distance was covered with a velocity $\mathrm{v}_{2}$ for half the time and with velocity $\mathrm{v}_{3}$ for the other half of time. The mean velocity averaged over the whole time of motion is\

Professor PC Thomas Classes · Answer

Let $S$ be the total distance travelled by body. If $t_{1}$ is the time taken to cover first one-third distance.\ Then $t_{1}=\frac{S / 3}{v_{1}}=\frac{S}{3 v_{1}}$\ Let $t_{2}$ be the time for each of the remaining two journeys.\ Then $\frac{2 S}{3}=v_{2} t_{2}+v_{3} t_{2}$ or, $t_{2}=\frac{2 S}{3\left(v_{2}+v_{3}ight)}$\ Take time $=\mathrm{t}_{1}+\mathrm{t}_{2}=\frac{\mathrm{S}}{3 \mathrm{v}_{1}}+\frac{2 \mathrm{~S}}{3\left(\mathrm{v}_{2}+\mathrm{v}_{3}ight)}$\ $=\frac{S}{3}\left[\frac{v_{2}+v_{3}+2 v_{1}}{v_{1}\left(v_{2}+v_{3}ight)}ight]$\ Average velocity\ $=\frac{	ext { total displacement }}{	ext { total time }}=\frac{\mathrm{S}}{\mathrm{t}_{1}+2 \mathrm{t}_{2}}$\