Average velocity is defined as total distance travelled divided by the total time needed to travel said distance.
You know that the body travels half of the total distance with a velocity of #v_0#. Let's say that it takes the body a total time of #t_(i)# to travel half of the total distance.
The other half of the total distance is travelled with #v_1# for half of the time, #t_1#, and with #v_2# for the other half of the time, #t_2#. Let's say that the total time it took the body to travel the second half of the total ditance is #t_(ii)#.
You know that
#"distance" = "velocity" * "time"#
You can use the fact that the two halves of the total distance are equal to write
#underbrace(v_0 * t_i)_(color(blue)("first half")) = underbrace(v_1 * t_1 + v_2 * t_2)_(color(blue)("second half"))#
But since #t_1 = t_2 = "t_(ii)/2#, you can write
#v_0 * t_i = v_1 * t_(ii)/2 + v_2 * t_(ii)/2#
#v_0 * t_i = t_(ii)/2 * (v_1 + v_2)# #" "color(blue)((1))#
This means that the average velocity can be written as
#bar(v) = overbrace(v_0*t_i + t_(ii)/2(v_1 + v_2))^(color(green)("total distance"))/underbrace((t_i + t_(ii)))_(color(red)("total time"))#
Use equation #color(blue)((1))# to replace #v_0 * t_i#
#bar(v) = (t_(ii)/2(v_1 + v_2) + t_(ii)/2(v_1 + v_2))/(t_i + t_(ii))#
#bar(v) = (t_(ii)(v_1 + v_2))/(t_i + t_(ii))#
Use equation #color(blue)((1))# again to express #t_i# in terms of the other parameters
#t_i = t_(ii)/(2v_0) * (v_1 + v_2)# #" "color(blue)((2))#
Use equation #color(blue)((2))# into the main equation to get
#bar(v)=(t_(Ii)(v_1 + v_2))/(t_(ii)/(2v_0) * (v_1 + v_2) + t_(ii))#
#bar(v) = (cancel(t_(ii))(v_1 + v_2))/(cancel(t_(ii))((v_1+v_2)/(2v_0) + 1))#
Thus,
#bar(v) = (v_1 + v_2)/((2v_0 + v_1 + v_2)/(2v_0)) = color(green)((2v_0(v_1 + v_2))/(2v_0 + v_1 + v_2))#
