Question

The average speed for a journey is the distance covered divided by the time taken [...] More in reply. Having issues with how to deduce and writing the equation for the second half (first half is solved)

Answer 1

One journey is completed by travelling for the first half of the time at speed \[v _{1}\]and the second half at speed \[v _{2}\]The average Speed these two are \[v _{a} = \frac{v _{1} + v _{2}}{2}\]

Another journey is complete by travelling at the speed \[v _{1}\]for half the distance and at speed \[v _{1}\]
Find the average speed \[v _{b}\]for the journey in terms of \[v _{1}\]and \[v _{2}\]Deduce that a journey is complete by travelling at two different speeds for equal distances will take longer than the same journey completed at the same two speeds for equal times.

Answer 2

In terms of finding the average speed of Vb:

d = 2(v1)(t1) = 2(v2)(t2)
t1 = d/(2v1), t2 = d/(2v2)

d = (va)(t1 + t2)
d = (va)[d/(2v1) + d/(2v2)]
1/(va) = 1/(2v1) + 1/(2v2)
va = 2(v1)(v2) / (v1 + v2)