Thread: Vectors help

Boat A has initial position (-8,9) and has a velocity vector [4,-1].
Boat B has initial position (1.-9).

Boat B has a maximum speed of 8km/h. Determine the time and velocity vector [a,b] for the collision to occur as soon as possible.

So hard..don't know how to do it.

There may be other ways.

Step 1. Let t be the time to the collision. Using a and b, write two equations on t. Dividing one equation by the other, you cat get a linear relation between a and b: they have to satisfy this relation in order for the boats to meet.

Step 2. Allowed points (a, b) are inside the circle with center (0, 0) and radius 8. The circle cuts a segment from the line obtained in step 1. Each point on the segment gives rise to some t; you need to find the point with the smallest t.

Step 3. From one of the equations from step 1 express t through a and draw a graph of t(a). Find the minimum of t when a belongs to the segment found in step 2.

Sorry, I still don't understand. I'm pretty dumb.

Let's take it one step at a time.

Originally Posted by emakarov

Let t be the time to the collision. Using a and b, write two equations on t.

If you can do this, write the first thing in the outline from post #2 that you are having trouble with.

What do you mean "write two equations on t"?

The values a, b and t are related by the fact that there is an encounter in t hours. In the horizontal direction, boat B, which moves right at a km/h, starts 1 - (-8) = 9 km to the right of boat A, which moves right at 4 km/h. The fact that they meet in t hours gives an equation that uses a and t. Similarly, there is an equation for the vertical direction.

Originally Posted by emakarov

The fact that they meet in t hours gives an equation that uses a and t.

Hmm...I don't know how to construct the equation.

The relative speed at which A catches up with B is 4 - a. There are 9 km to catch up in t hours.

Alternatively, boat A will be at x-coordinate -8 + 4t in t hours. Similarly, boat B will be at 1 + at. These coordinates coincide when they meet.

Originally Posted by emakarov

The relative speed at which A catches up with B is 4 - a. There are 9 km to catch up in t hours.

Alternatively, boat A will be at x-coordinate -8 + 4t in t hours. Similarly, boat B will be at 1 + at. These coordinates coincide when they meet.

So the y-coordinate for boat A would be 9-t and for boat B would be -9+bt ?

Yes.

Okay, so now do I equate them?

I am sure they won't blow up if you do

Originally Posted by emakarov

I am sure they won't blow up if you do

I got: a=(-9/t)+4 and b=(18/t)-1

My original idea was to eliminate t and to express b through a.

Originally Posted by emakarov

My original idea was to eliminate t and to express b through a.

How do I eliminate t?