Ship A is 10 km due west of ship b . Ship A is heading directly north at a speed of 30 km/h, while ship b is heading in a direction `60^(@)` west of north at a speed of 20 km/h .
(i) Determine the magnitude of the velocity of ship B relative to ship A .
(ii) What will be their distance of closest approach ?

Explanation

Velocity of A = 30 kmph due North `:. V_(A) = 30 hat(j)`
Velocity of b = 20 kmph `60^(@)` west of North
`:. V_(B) = - 20 sin 60^(@) + 20 cos 60^(@) = 10 sqrt(3) hat (i) + 10 hat(j)`
Velocity of b w.r.t A `= V_(B) - V_(A)`
`= - 10 sqrt(3)hat(i) + 10 hat (j) - 30 hat(j) = - 10 sqrt(3)hat (i) - 20 hat(j)`
`V_(R) = | V_(B) - V_(A)| = sqrt(100 xx 3 + 400) = 10 sqrt(7)` kmph .
Shortest distance : In `Delta le ` ANB shortest distance , AN = AB sin `theta`
But distance, AB = 10 km
`:. AN = 10 xx (20)/(10sqrt(7)) = (20)/(sqrt(7)) = 7 . 56 km `