Boat Collide Problem

This page is still under construction. Any ideas for specific questions are most welcome: LFS

Problem setting: Two ships are sailing in the fog and are being monitored by tracing equipment. As they come into the observer's rectangular radar screen, one ship, the Rusty Tube, is at a point 900 mm to the right of the bottom left corner of the radar screen along the lower edge. The other ship, the Bucket of Bolts, is located at a point 100 mm above the lower left corner of that screen. One minute later, both ships' positions have changed. The Rusty Tube has moved to a position on the screen 3 mm left and 2 mm above its previous position on the radar screen. Meanwhile, the Bucket of Bolts has moved to a position 4 mm right and 1 mm above its previous location on that screen.

Question: Assume that both ships continue to move at a constant speed on their respective linear courses. Using graphs and equations, find out if the two ship will collide.

Animated simulator

To animate, click on the play button at bottom left of graph.

To animate manually, right-click on slider and deselect "Animation on". Then, click and drag the point on the slider.

Building the simulator step-by-step Notes on directions

Notes on 3 and 4: Using the input window to input A and B makes these points movable off the axes. (Clicking to define makes them points on the axes.)
Notes on 5 and 6: Notice the names At1 and Bt1. Don't use object names A1, A2,..., B1,B2, ..., etc. These names are now reserved for spreadsheet entries.
Notes on 7 and 8: Of course you can use the line tool. But A and At1 are very close together on this big grid, so inputting is probably easier.

The graph
1. Open GeoGebra (http://geogebra.org)
2. Change the graphics view to fit the 2 starting points.
2a. Put (0,0) down by the bottom left corner and zoom-out so that the window is (1100,800).
2b. Turn the grid on. (Help is coming for this.)
3. Input point A. Click in input window and type: A=(900,0) and hit Enter.
4. Input point B. Type: B=(0,100) and hit Enter.
5. Input point At1. Type: At1=A+(-3,2) and hit Enter.
6. Input point Bt1. Type Bt1=B+(4,1) and hit Enter.
7. Input line a. Type line[A,At1] and hit Enter.
8. Input line b. Type line[B,Bt1] and hit Enter.

The animated simulator
9. Input number t. Type t=0 and hit Enter.
10. Make t a slider. Right-click on t in algebra window. Select "Properties". Click & drag the properties box down a bit.
  10.a On the Basic tab, select "Show object". (Slider should appear in graphics window.) and       check that "Show label" is selected.
  10.b On the Slider tab: min=0, max=200, increment=1, width=400, repeat=increasing
  10.c Click on Close.
11. Input point At. Click in input window and type: At=A+t*(-3,2) and hit Enter.
12. Input point Bt. Type Bt=B+t*(4,1) and hit Enter.

Right-click on the slider t and select: Animation on.

Why I like this question and some good questions.

The graph looks like every 2x2 system of linear equations they have solved in Algebra 1.
- It looks like the boats collide at the intersection point.
- It seems like all they need to do is solve the system and be done.
- ... until you say "Where is time on the graph?".
The student can build a animated simulator that "shows time" - easily!
- Then they can see that the boats do not collide.
- Below is a simulator I built using the freeware GeoGebra.
The kids can make the boats collide - what fun!.
- They can move the starting points until they get the boats to collide.
- They can also adapt the simulator so that they can change the slopes and get the boats to collide. Directions here.
- My thanks to David Cox for seeing this!
You can get all kinds of mathematics out of them.
- Have them calculate when each of the boats reaches the intersection point in the original question (see below).
- Have them check the math on their "colliding simulator" to see if the boats really do collide, where and when.
- Ask them about a 3D graph and what this would look like when the boats don't collide and when they do.

Algebraic solution to original question

Rusty Tube's starting co-ordinates are (900,0).
Moves by (-3,+2) so the slope (gradient) is \frac{\,\,2}{-3} = -0.6666
This means the equation is y=-0.6666x +c .
Substituting y=0 and x=900 , 0=-600+c or c=600 .
So the equation of RT's journey is y= -0.6666x+600 .

Bucket of Bolts starting co-ordinates are (0,100).
Moves by (+4,+1) so the slope (gradient) is \frac{1}{4} = 0.25 .
Which means the equation is y=0.25x +c .
Substituting y=100 and x=0 , 100=0+c or c=100 .
So the equation of BB's journey is y= 0.25x+100 .

Put the equations together and work out where on the x-axis their journeys will cross.
That is, solve 2x2 system of linear equations: \left\{ \begin{array}{l}y= -0.6666x+600 \\y= 0.25x+100 \\ \end{array} \right. .
  Set them equal and solve for x:
   -0.6666x +600=0.25x+100
      500=0.91666x
         x=500/0.91666
         x=545.45

We do not need the value of y at this point for the problem, but it is:
y= -0.6666x+600 where x=545.45 . So: y=236.37

Now find the time taken for each boat to get there and if they are the same they will collide.

RT starts at 900 and travels horizontally x=-3 each minute, so 545.45=900-3t
Solving for t: t= 354.545 / 3 or t = 118.18 [min]

BB starts at 0 and travels horizontally x=+4 each minute, so 545.45=0+4t
Solving for t: t= 545.45 / 4 or t = 136.36 [min]

Since 118.18 \ne 136.36 , the boats will not collide.

Answer adapted from Dan M

One answer to getting the boats to collide: