How do I show that 2 moving objects collide?

[SueJ]

Homework Statement

We have a boat sailing at a uniform acceleration of (4i, +2j) from the origin, and a branch floating at a uniform velocity of (-4i, +j) starting from position (16i, +2j). Show that they collide, and give the time of the collision

Relevant Equations

1. d=ut+1/2(at^2)
2. v^2=u^2+2ad
3. v=u+at
4. d=t(u+v)/2

They collide when their positions are the same, so I plugged the information for the boat into equation 1 to get an expression for d which is (2i, +j)t^2

Then I used equation 4 to get an expression for d for the branch, which is (-4i, +j)t

I would need to take into account the different starting position for the branch by subtracting it from the branch side

So they collide when (2i, +j)t^2 = (-4i, +j)t - )16i, +2j)

I then tried to factorise it, and this is where I got stuck!

 

[haruspex]

by subtracting it from the branch side

Have another think about that.

I then tried to factorise

Don’t forget that a 2D vector equation is really two scalar equations. Try writing them separately.

 

[SueJ]

yep, you are right, thanks - ADD it to the branch side.
I did try separating the i and the j so I ended up with 2 quadratics - the one for i did factorise fine, the j one didnt - I may have made a mistake obviously - did you try it, and if so, did it factorise?
In order to SHOW that they collide, I just drew a graph of the postion of boat and branch at increasing t just to show that they cross - is this what would be expected for the question 'show that they collide'?

 

[SueJ]

Have another think about that.

Don’t forget that a 2D vector equation is really two scalar equations. Try writing them separately.

oh yes I did make a mistake and I got the same t for both I and j this time, thanks