Two parallel rail tracks run north-south. Train A moves north with a speed of `54 km h^(-1)`, and train B moves south with a speed of `90 km h^(-1)`. What is the
(a) velocity of B with respect to A ?,
(b) velocity of ground with respect to B ?, and
(c) velocity of a monkey running on the roof of the train A against its motion (with a velocity of `18 km h^(-1)` with respect to the train A) as observed by a man standing on the ground ?

Explanation

If we consider the direction from South to North as positive X-axis then,
`v _(A) = + 54 kmh^(-1) and v _(B) =- 90 kmh^(-1)`
`- (54 xx 1000)/(3600) = - (90 xx 1000)/(3600)`
`= 12 ms ^(-1) =- 25 ms ^(-1)`
Velocity of B w.r.t `A = v _(B) - v _(A)`
`=- 25 -15`
`=- 14 ms ^(-1)`
`therefore` The train B appears to A to move with a speed of `40 ms ^(-1)` from north to south.
(b) Velocity of B w.r.t ground `= v _(G) - v _(B) = 0- (-25)`
`=25 ms ^(-1)`
(c ) Suppose `v _(M)` is the velocity of monkey w.r.t. ground,
`therefore` Velocity of monket w.r.t A
`u_(MA) =- 18 kmh ^(-1)`
`= - (18 xx 1000)/( 3600)=- 5 ms ^(-1)`
But `v _(MA) = v _(M) - v _(A)`
`therefore v _(M) = v _(MA) +v _(A)`
`=- 5 + 15`
`=10 ms ^(-1)`