Question

P and Q run around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after Q has travelled 80 metres; and meet a second time 40 metres before P has completed one lap, find the circumference of the track

Answer 1

let P and Q meet intially at time t
then their respective speed s are V(p)=80/t and V(q)=pi*r-80/t (q traveled distance of half circumference-80 units)
now they again meet . at this instance equating time =distance /speed
2pi*r-40/80/t = 2pi*r+40/pi*r-80/t
cross multiply
160pi*r+3200=2* (pi*r)^2-40 p*r - 160 pi*r -3200
let 2pi *r =x(circumference to be determined )
then 80x+3200=x^2-20x-80x-3200
then solve the quadratic to get x=210.416 units
PS: there may be calculation errors

Answer 2

Let C be circumference around track
Let t be time it takes when they 1st meet
Let P and Q represent the respective velocities

\[P*t = \frac{C}{2}-80 .............. Q*t =80\]
\[\rightarrow P = \frac{C-160}{2t} .............Q =\frac{80}{t}\]
when they meet 2nd time, setting times equal
\[\frac{C-40}{P} = \frac{C/2 +40}{Q}\]
\[\rightarrow \frac{2t(C-40)}{C-160} = \frac{t(C+80)}{160}\]
dividing by t cancels it out
cross-multiply and solve for C
\[C = 400\]