Vector Paths and Initial Positions: Boat Collision Question Solved"

[danago]

Hey. Heres the question:

Two boats are moving along straight paths and their position vectors at noon are:

$$\mathbf{r}_1=(7-4t)\mathbf{i}+(-5+t)\mathbf{j}$$
$$\mathbf{r}_1=(12-3t)\mathbf{i}+(13-t)\mathbf{j}$$

a) where was the first boat initially?

b) Where was the second boat relative to the first boat initially?

c) What is the velocity vector, $$\mathbf{v}_2$$, of the second boat?

d) find weather or not the boats will colide.

For the first question, i assumed its initial position would be 0 hours after noon. So i just answered it as:

$$7\mathbf{i}-5\mathbf{j}$$

Now i wouldn't have a clue if that's even close to being correct, but its the only decent answer i could come up with.

For the next part, i drew the diagram, and just found a vector going from the position of the first boat to the second, from the initial positions, which gave me the final vector:

$$4\mathbf{i}+18\mathbf{j}$$

For part c, the velocity of the second boat, i just wrote how much the position vector increases for every incriment of t. I came up with:

$$-3\mathbf{i}-\mathbf{j}$$

The problem with this was that the question said they were traveling in a straight line, and if i apply this velocity, then they change their direction. So I am lost.

And with the final question, I am stuck, because i can't really do it until i answer the previous ones correctly.

So if anyone doesn't mind, please put me on the right track for these questions, because i highly boubt I've answered them correctly.

Thanks,
Dan.

 

[neutrino]

You're first and third answers are right. The second should be 5i + 13j

The problem with this was that the question said they were traveling in a straight line, and if i apply this velocity, then they change their direction. So I am lost.

I don't see a problem there. It's a constant vector.

And for the last part, what's the relation between the position vectors of the boats when they collide?

 

[danago]

You're first and third answers are right. The second should be 5i + 13j

I don't see a problem there. It's a constant vector.

And for the last part, what's the relation between the position vectors of the boats when they collide?

could you explain how you got 5i + 13j please? I re did it, and ended with the i component being 5, but i don't understand how you got 13 for the j component.

And ill try the final question now.

 

[neutrino]

could you explain how you got 5i + 13j please? I re did it, and ended with the i component being 5, but i don't understand how you got 13 for the j component.

And ill try the final question now.

Sorry, that was a typo.

 

[danago]

so its 5i + 18j then?

Anyway, I am doing the final question now. So to find when they collide, i need to find then their position vectors are the same, at the same value for t.

So do i just equate the components for each vector?:
i 7-4t
j -5+t

i 12-3t
j 13-t

So then i need to find when the i and j components of both ships are the same. I got that when the i component of both boats is 27, the j component will be 4, so they will collide at:

$$27\mathbf{i}+5\mathbf{j}$$

Im not sure if that's right :S

 

[Hootenanny]

I found that the boats do not collide.

-Hoot

 

[Hootenanny]

You can check if your answer is correct (whether they collide at your defined point), choose a vector equation and sub your values in for i and j such as this;

$$r_{1} = (7-4t)j + (t-5)j$$

Subbing $27i + 5j$ into each component;

$$7 - 4t = 27$$

$$t - 5 = 5$$

Now, do they both return the same value of t?

-Hoot

 

[danago]

yea i thought about it again, and no, they don't give the same values of t. Thanks for the help everyone :)