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# Distance of closest approach of two ships problem

Two ships are 10 km apart on a line running south to north. The one farther north is steaming west at 20 km/h. The other is steaming north at 20 km/h. What is their distance of closest approach? How long do they take to reach it?

The main hindrance to solving problem is that students fail to visualize the situation.

Let’s represent the condition using a diagram. Let the first ship steaming north be named A and the other B. The relative velocity of A with respect to B is 20 √2 south west.  If we extend the line along the 45° south west from the initial position of A, the distance of closest approach is the perpendicular drawn from B to this line.

So, the distance of closest appraoch = 10 sin 45° = 10/√2 = 7.07 m

Time taken to reach the distance of closest approach = distance of closest approach / relative speed

= (10/√2)/(20 √2)

= 0.25 h

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