# Another one on ships

#### uxinox

Hi guys,

I have used the sledgehammer to crack this one, but have no idea how to do it elegantly!

A ship A is moving eastwards with a speed of 15 km/h and another ship B, at a given instant 10 km east of A, is moving southwards with a speed of 20 km/h. How long after this instant will the ships be nearest to each other, how far apart will they be then, and in what direction will B be sighted from A?

The relative vector didn't tell me much, so I started plotting what happened after one minute then 10, 30 up to 40 when both ships are aligned north south. After 10 minutes, for example, vector AB is the hypotenuse of a triangle 9.5 km (east) and 3.3 km (south) giving a distance between the ships of 8.194 km.

so after 0 |AB| = 10 km
1 |AB| = 9.76
10 = 8.19
20 = 8.34
40 = 13.3

from this I guessed the min distance would be between 1 and 10 minutes and I could have plotted all distances for each minute. But I don't believe this is the way to do it!
Any better suggestions would be appreciated!

#### oz93666

One way to do this is to define A as being stationary .... then the velocity of B (relative to A) is 20 km/hr south plus 15 km/hr west ... another 3:4:5 triangle ...

the speed of B is 25 km/hr moving in the direction given by the triangle , sort of south westish (not exactly)

Now for the distance ... put a dot on the paper, this is A put another dot to the right of A a distance 10 away ... this is B , draw a line from B to left hand corner at the correct angle for a 345 triangle this is the path of B relative to A , geometry can get the distance and times

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#### uxinox

thanks oz.

I did it along the lines you suggested, but got my relative vector of speed 25 km/h going 37 degrees N of E. BUT ... the question asks for the time of min distance between A and B, what this distance is and the direction of B from A.

I don't see how to get this information from the relative vector.

The direction is given as 37 degrees South of East. I get 37 degrees North of East and you had South of West!

#### oz93666

so if you put a dot on the paper this is A ...measure 10 units along in direction East and put a dot for B ...this is the distance apart , then B has to be moving towards the left corner (SW ish) ...

If B were moving SE ish or NE ish (relative to A) this would take it away from A so there would be no minimum distance.

So you have the line from B moving down and to the left , you know it's angle ... now project a right angle from this line to hit A ... this is the minimum distance , which you can find since you have one side , 10Km , and the angle .. from this distance you can calculate the time , since you know the speed .

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#### THERMO Spoken Here

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#### uxinox

Thanks oz,

I see what you mean.