A point traversed half the distance with a velocity v0. The remaining part of the distance was covered with velocity v1 for half the time, and with velocity v2 for the other half of the time. Find the mean velocity of the point averaged over the whole time of motion.

Solution:

Let s be the total distancetraversed by the point and t1 the time taken to cover half the distance. Further let 2t be the time to cover the rest half of the distance.

Therefore  \( \frac{s}{2}={{v}_{0}}{{t}_{1}}\,\,\,\,\,\,\,\,\,or\,\,\,\,\,\,\,\,{{t}_{1}}=\frac{s}{2{{v}_{0}}}\,\,\,\,\,\,\,\,\,(1) \)

And  \( \frac{s}{2}=({{v}_{1}}+{{v}_{2}})t\,\,\,\,\,\,\,\,or\,\,\,\,\,\,2t=\frac{s}{{{v}_{1}}+{{v}_{2}}}\,\,\,\,\,\,\,\,\,\,\,(2) \)

Hence the sought average velocity

 \( <v>=\frac{s}{{{t}_{1}}+2t}=\frac{s}{\left( \frac{s}{2{{v}_{0}}} \right)+\left( \frac{s}{{{v}_{1}}+{{v}_{2}}}\right)}=\frac{2{{v}_{0}}({{v}_{1}}+{{v}_{2}})}{{{v}_{1}}+{{v}_{2}}+2{{v}_{0}}} \)

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