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 v₀. Let the time taken be t₀ for this. Hence,

d / 2 = v₀ * t₀
t₀ = d / (2 * v₀)

Now, for the second half of the total distance travelled, the point travelled at v₁ for the first t₁ seconds and at v₂ for the next t₁ seconds. Hence,

d / 2 = v₁ * t₁ + v₂ * t₁
t₁ = d / (2 * (v₁ + v₂))

Average speed = Total Distance / Total Time Taken
Hence,

v_avg = d / (t₀ + t₁ + t₁)
v_avg = d / (d / (2 * v₀) + d / (v₁ + v₂))
v_avg = 1 / (1 / (2 * v₀) + 1 / (v₁ + v₂))
v_avg = 1 / ((2v₀ + v₁ + v₂) / (2 * v₀ * (v₁ + v₂)))
v_avg = (2 * v₀ * (v₁ + v₂)) / (2v₀ + v₁ + v₂)
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