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Ship A is sailing towards north-east wit...

Ship `A` is sailing towards north-east with velocity `vecr=30hati+50hatJ km//hr` where `hati` points east and `hatj`, north. Ship `B` is at a distance of `80km` east and `150km` norht of Ship `A` and is sailing towards west at `10km//hr`. A will be at minimum distance from `B` in:

A

2.2 hrs.

B

3.2 hrs.

C

4.2 hrs.

D

2.6 hrs.

Text Solution

Verified by Experts

The correct Answer is:
D

Position of B as function of time `= 80 hati + 150 hatj - 10t hati`
Position of A as function of time `30hati + 50t hatj`
`:. Relative position of B as fraction of time = `80hati + 150 hatj - 40t hati - 50t hatj`
Taking magnitude square `=(80 - 40t)^(2) + (150 - 50t)^(2)`
Time derivative = `2(80 -40t)(-40) + 2(150 - 50t) (-50) = 0`
`implies (8-4t)(-4) + (15-5t) (-5) = 0" "implies "16t + 25t = 32 + 75`
`implies 41t = 107 " "implies" " t = 107/41 = 2.6` hrs.
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