# Nearest Distance between 2 boats

• zimbabwe
In summary: Remember that the distance of closest approach will be the same along any line perpendicular to the relative velocity vector.
zimbabwe
Ship A is 10 km due west of ship B. Ship A is heading directly north at a speed of 30km/hr, whilst ship B is heading in a direction 60 degress west of norht at a speed of 20km/hr.

a) Deteremine the magnitude and direction of the velocity of ship B relative to ship A.

b) What will be their distance of closest approach?

My Attempt for the first part

B Relative to A, first into compentents, so sin(60)*20=17.3 and cos(60)*20=10
Then relative to A
would be 10 km west and 30-17.3= 12.67 km south

however the actual answer is Velocity is 26.5 km/h in a direction South 41 degrees West.

How was i supposed to do it, and how do i start part b.

zimbabwe said:
Ship A is 10 km due west of ship B. Ship A is heading directly north at a speed of 30km/hr, whilst ship B is heading in a direction 60 degress west of norht at a speed of 20km/hr.

a) Deteremine the magnitude and direction of the velocity of ship B relative to ship A.

b) What will be their distance of closest approach?

My Attempt for the first part

B Relative to A, first into compentents, so sin(60)*20=17.3 and cos(60)*20=10
Then relative to A
would be 10 km west and 30-17.3= 12.67 km south

however the actual answer is Velocity is 26.5 km/h in a direction South 41 degrees West.

How was i supposed to do it, and how do i start part b.

For part a), you have simply got the components confused. Sin() gives the 'west' component not 'south', and vice versa. Try drawing the triangle to see your mistake (sin = opp/hyp, etc).

For part b), now that you have the relative velocity vector you can set up t-dependent vector R giving the relative position of the boats. Now the problem is simply a matter of finding the shortest distance between a point and a line. One way to do it is to construct a line which goes through the origin and whose dot-product with R is zero (therefore orthogonal to R). Then evaluate the point of intersection of the two lines and take the magnitude of the separation.

Right i see part the mistake, so now i have 20 km/hr south and 17.3km/hr west. Find the hypontuse of this I get 26.4 km/hr, ok i got it. Thank you

b)i don't see it, by vector r does it mean, r=(10,0)+t(-17.3,-20)

zimbabwe said:
Right i see part the mistake, so now i have 20 km/hr south and 17.3km/hr west. Find the hypontuse of this I get 26.4 km/hr, ok i got it. Thank you

b)i don't see it, by vector r does it mean, r=(10,0)+t(-17.3,-20)

Right, R gives the equation of a line in the x-y plane, with direction (-17.3,-20). So a vector perpendicular to the line could be

L = m(20,-17.3)

With m a parameter which can take any value (similarly to t).

We then find the points (x,y) where the two lines cross:

x = 20m = 10 -17.3t
y = -17.3m = -20t

Now solve the simultaneous equations => t=0.25 etc

A quicker way to do it is to spot that the point we want occurs at

R (dot) [direction of line] = 0

i.e. 10*-17.3 + t(-17.3^2 + -20^2) = 0

=> t = 0.25 etc

A third way to do it is to simply take the modulus of R and minimise it w.r.t. t. Takes a bit more algebra but requires less geometrical thinking.

## 1. What is the nearest distance between 2 boats?

The nearest distance between 2 boats is the shortest distance between their closest points. This can be measured in meters or nautical miles.

## 2. How is the nearest distance between 2 boats calculated?

The nearest distance between 2 boats can be calculated using the Pythagorean theorem, which states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In this case, the hypotenuse represents the nearest distance between the boats, while the other two sides represent the distance between each boat and a reference point.

## 3. Why is it important to know the nearest distance between 2 boats?

Knowing the nearest distance between 2 boats is important for safety reasons. It helps prevent collisions and allows boats to maintain a safe distance from each other while navigating. It is also important for determining the right of way and avoiding potential hazards such as shallow waters.

## 4. Can the nearest distance between 2 boats change?

Yes, the nearest distance between 2 boats can change due to various factors such as the speed and direction of the boats, weather conditions, and navigational decisions. It is important for boat operators to constantly monitor and adjust their distance from other boats to ensure safety.

## 5. How can technology help in determining the nearest distance between 2 boats?

Technology such as radar, sonar, and GPS can provide real-time information on the distance between boats and their locations. This can help boat operators make informed decisions and avoid potential collisions. Additionally, AIS (Automatic Identification System) technology allows boats to communicate with each other and exchange information on their positions and movements, further enhancing safety and navigation.

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