A man starts walking on a circular track of radius `R`. First half of the distance he walks with speed `V_(1)` , half of the remaining distance with speed `V_(2)`, then half of the remaining time with `V_(1)` and rest with `V_(2)` and completes the circle. Average speed of the man during entire motion in which he completes the circle is.

Options

A. `(2V_(1)V_(2)(V_(1)+V_(2)))/(V_(2)^(2)+2V_(1)^(2)+2V_(1)V_(2))`
B. `(4V_(1)V_(2)(V_(1)+V_(2)))/(V_(1)^(2)+2V_(2)^(2)+5V_(1)V_(2))`
C. `(V_(1)V_(2)(V_(1)+2V_(2)))/(V_(1)^(2)+V_(2)^(2)+4V_(1)V_(2))`
D. `((V_(1)+2V_(2))^(2))/(V_(1)+V_(2)+2V_(1)V_(2)^(2))`

Answer

B

Explanation

Let total distance travelled is 4s
`2s rarr V_(1 rarr t_(1)) = (2s)/(V_(1))`
`s rarrV_(2) rarr t_(2) =(s)/(V_(2))s[{:(V_(1) rarr t_(0)),(V_(2) rarr t_(0)):}`
`(V_(1)+V_(2))t_(0)=s rArr t_(0)=(s)/(V_(1)+V_(2))`
`lt V gt=(4s)/(t_(1)+t_(2)+2t_(0))=(4s)/((2s)/(V_(1))+(s)/(V_(2))+(2s)/(V_(1)+V_(2)))`
`=(4 V_(1)V_(2)(V_(1)+V_(2)))/(2V_(2)(V_(1)+V_(2))+V_(1)(V_(1)+V_(2))+2V_(1)V_(2))`
`=(4 V_(1)V_(2)(V_(1)+V_(2)))/(2V_(1)V_(2)+2V_(2)^(2)+V_(1)^(2)+V_(1)V_(2)+2V_(1)V_(2))`
`=(4V_(1)V_(2)(V_(1)+V_(2)))/(V_(1)^(2)+2V_(2)^(2)+5V_(1)V_(2))`