### Problem 1.2

Sreepad had gone early to the office today owing to some meetings. Hence, Sanjeev and I were driving down together. This problem was a pretty straight forward one and we solved this independently without much ado.

Solution:

Let the total distance travelled be d. The velocity of travel for the first d/2 km is v

_{0}. Let the time taken be t

_{0}for this. Hence,

d / 2 = v

_{0}* t

_{0}

t

_{0}= d / (2 * v

_{0})

Now, for the second half of the total distance travelled, the point travelled at v_{1} for the first t_{1} seconds and at v_{2} for the next t_{1} seconds. Hence,

_{1}* t

_{1}+ v

_{2}* t

_{1}

t

_{1}= d / (2 * (v

_{1}+ v

_{2}))

Average speed = Total Distance / Total Time Taken

Hence,

v

_{avg}= d / (t

_{0}+ t

_{1}+ t

_{1})

v

_{avg}= d / (d / (2 * v

_{0}) + d / (v

_{1}+ v

_{2}))

v

_{avg}= 1 / (1 / (2 * v

_{0}) + 1 / (v

_{1}+ v

_{2}))

v

_{avg}= 1 / ((2v

_{0}+ v

_{1}+ v

_{2}) / (2 * v

_{0}* (v

_{1}+ v

_{2})))

v

_{avg}= (2 * v

_{0}* (v

_{1}+ v

_{2})) / (2v

_{0}+ v

_{1}+ v

_{2})

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