Math/Physics: Given angle and vector find point of intersection?

0
votes

I have two objects and I need to determine whether they will collide at some time in the future. Say the two objects are trucks, each in their separate lane. The first truck wants to change lanes, but will the trucks collide?

Both objects are moving forward, but only the first has an angle. The objects are moving at different, varying velocities, yet only the velocity of the second object is known. Also, the two objects are operating on separate coordinate systems (meaning each object has a new origin), with varying distance between the objects.

I don't have a vector for the first object, but I do have an angle (not fixed) at which it is headed, relative to the x-axis.

For the second object, I have a vector, and it is headed straight.

Please refer to the picture below:

Collision

How can I find the point of intersection, based on these criteria? I imagine the angle is an infinite line, which at some point will cross the vector magnitude of the second object. I don't suppose this is as straight forward as calculating the point of intersection of two lines.

Wrapup:

  • Velocity of the second object is known (but varying)
  • Angle of the first object is known
  • Distance between the objects is known (but varying)
  • Acceleration in the x-direction of the first object is known (But I CANNOT use this to find the velocity due to limitations)
1
vectorcollision-detectionintersectionangle
Because you don't know the velocities of the objects, you can't guarantee that they'll collide. You could approximate where they potentially collide, but again, without velocities, you'll have no certainties. - Mr. Llama
Do you at least know the position of the objects relative to each other? - Mr. Llama
I forgot to mention that. I have the distance between the two objects. - Rex
Distance and angle, or just distance? Could you possibly give us a bit of context for your question? Right now it seems rather abstract. - Mr. Llama
I realize that. I just updated the question. - Rex

1 Answers

5
votes

I have redrawn your image with a few more labels on it.

First we need to be working in one coordinate system. I have chosen to use the system with object 1 at (0,0). This means that object 2 is at position (Xd,Yd).

Then we find the point of intersection. In these coordinates this point will be x=0 and y=y1 (0,y1). Now we want to know what this is in terms of y2, yd and the angle theta.

eq1

Using what you know this now tells us where the intersection point is. We need to find out how long it takes each object to get to that point. Taking this time for each and setting them equal to each other will give us the conditions on the velocity and acceleration of each object that will make them collide. Starting with the constant acceleration equation for object 1 and 2 (find these in any intro physics book):

ca1

ca2

where y0 is the starting position, V0 is the initial velocity, and a is the acceleration.

y0 for object 1 will be 0 since we called that the origin and y0 for object 2 will just be yd.

Now solve for t in each of these and set them equal to each other. Since these are just quadratic in t we can use the quadratic equation. After this you should get:

full

with,

with

Then plug in the values you know for velocities, accelerations and positions. If both sides are equal they collide, if not, they dont. This essentially sets conditions on these quantities that have to be met in order for them to be at the same place, at the same time, i.e. a collision.