From two stations A and B, two trains started simultaneously towards each other with velocities `v_(1) "and" v_(2)` respectively. After they crossed each other the first train reached B in time `t_(1)` and the second train reached A in time `t_(2)`. Show that `v_(1) : v_(2) = `sqrt(t_(2)):sqrt(t_(1))`

Explanation

Let the two trains crossed each other at point O after time t. So,
AO = `v_(1)` t and BO = `v_(2) t i.e, (AO)/(BO) = (v_(1))/(v_(2))`
Again for the first train OB = `v_(1)t_(1)`
and for the second train OA = `v_(2)t_(2)`
`:. " " (OA)/(OB)=(AO)/(BO) =(v_(2)t_(2))/(v_(1)t_(1))`
Then `(v_(1))/(v_(2)) = (v_(2)t_(2))/(v_(1)t_(1)) " ""or," (v_(1)^(2))/(v_(2)^(2)) = (t_(2))/(t_(1)) " ""or,"(v_(1))/(v_(2))=(sqrt(t_(2)))/(sqrt(t_(1)))`