Question

This is the question with the options

Answer 1

(c)
`(2v_0(v_1+v_2))/(v_1+v_2+2v_0)`

Step-by-step explanation:

Average speed is calculated as (total distance)/(total time). We have three segments in the journey, indexed by 0, 1, and 2:

`v = (s)/(t)=(v_0t_0+v_1t_1+v_2t_2)/(t_0+t_1+t_2)\\t_1=t_2\implies\\v=(v_0t_0+v_1t_1+v_2t_1)/(t_0+t_1+t_1)=(v_0t_0+v_1t_1+v_2t_1)/((v_0t_0)/(v_0)+(v_1t_1)/(v_1)+(v_2t_1)/(v_2))`

We also know that the distance of the first segment is the same as one of segment 2 and 3 together:

`v_0t_0=v_1t_1+v_2t_1\\v_0(t_0)/(t_1)=v_1+v_2`

Going back to the average speed expression, divide by t_1:

`(v_0t_0+v_1t_1+v_2t_1)/((v_0t_0)/(v_0)+(v_1t_1)/(v_1)+(v_2t_1)/(v_2))=((v_0t_0)/(t_1)+v_1+v_2)/((v_0t_0)/(v_0t_1)+2)`

and combine the two equations:

`((v_0t_0)/(t_1)+v_1+v_2)/((v_0t_0)/(v_0t_1)+2)=(2(v_1+v_2))/((v_1+v_2)/(v_0)+2) = (2v_0(v_1+v_2))/(v_1+v_2+2v_0)`

The last form matches your choice (c).