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For the first leg of the journey,
V = 0 + α  t1
So  V = α  t1

for the second leg of journey of duration  t - t1 :
0 = V - β (t - t1)
0 = α t1 - β t + β t1

t1 = β t / (α + β)

V =  αt1 = α β t / (α + β )

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