# A ship A is moving Westwards with a speed of 10 km h -1 and a ship B 100 km South of A, is moving Northwards with a speed of 10 km h-1. The time after which the distance between them becomes shortest, is: Option 1) $5\sqrt2\: \: h$ Option 2) $10\sqrt2\: \: h$ Option 3) $0\: \: h$ Option 4) 5 h

Answers (1)

As we learnt in

Case of Relative velocity -

When A & B are moving along with straight line in opposite direction.

$\vec{V}_{A}=$ Velocity of object A.

$\vec{V}_{B}=$ Velocity of object B.

Relative velocity of A with respect to B is.

$\vec{V}_{AB}=\vec{V}_{A}-(-\vec{V}_{B})$

$-\vec{V}_{B}$  is because of opposite direction.

$\vec{V}_{AB}=\vec{V}_{A}+(\vec{V}_{B})$

- wherein

E.g A body moves in east with speed of 20m-1, And body in west with velocity of 5ms-1.

Find velocity of A with respect to B.

$\vec{V}_{AB}=\vec{V}_{A}+(\vec{V}_{B})$

= (20+5 ) ms-1

= 25 ms-1

$\\sin 45=\frac{PQ}{OQ}\\ PQ=100\times \frac{1}{\sqrt{2}}=50\sqrt{2}m\\ V_{AB}=\sqrt{V_{A}^{2}+V_{B}^{2}}=\sqrt{10^{2}+10^{2}}=10\sqrt{2}$

Time taken to reach shortest path is

$t=\frac{PQ}{V_{AB}}=\frac{50\sqrt{2}}{10\sqrt{2}}=5h$

Correct option is 4.

Option 1)

$5\sqrt2\: \: h$

Incorrect

Option 2)

$10\sqrt2\: \: h$

Incorrect

Option 3)

$0\: \: h$

Incorrect

Option 4)

5 h

Correct

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