# I Derivation of motion equations for collision detection

1. May 18, 2017

### M Omran

Dear all,

I am working on a car-accident prediction model. I found the following equations here that worked in my model. However, I could not figure out how the equations were derived. Could any one help me in understanding the derivation of the mintime and mindist equations (Steps 3 and 5 below).

1. For each car/object/point, you know the following:

initial x position
speed in x direction
initial y position
speed in y direction

2. For two points a and b then, lets define the following variables:

xa0: initial x position of point A
xat: speed in x direction of point A
ya0: initial y position of point A
yat: speed in y direction of point A

xb0 initial x position of point B
xbt speed in x direction of point B
yb0 initial y position of point B
ybt speed in y direction of point B

3. The distance between to points is given by Pythagoras. By expressing this distance as a function of time (based on the given variables above), then we can solve for the minimum possible distance. If we do this then we get a formula which will give us the time of minimum distance. The result is the following formula:

mintime =
-(xa0*xat - xat*xb0 - (xa0 - xb0)*xbt + ya0*yat - yat*yb0 - (ya0 - yb0)*ybt)
/
(xat^2 - 2*xat*xbt + xbt^2 + yat^2 - 2*yat*ybt + ybt^2)

4. The distance at any time t is given by the following expression:

dist = sqrt((t*xat - t*xbt + xa0 - xb0)^2 + (t*yat - t*ybt + ya0 - yb0)^2)

It depends on the given variables in addition to t.

5. To calculate the minimum distance then, you just evaluate it using the minimum time you calculated previously. So with t = mintime:

mindist = sqrt(
(mintime*xat - mintime*xbt + xa0 - xb0)^2
+
(mintime*yat - mintime*ybt + ya0 - yb0)^2
)

2. May 18, 2017

### Staff: Mentor

(4) is clear?
Calculate its time-derivative. At the minimal distance, the derivative is zero. That gives you an equation that can be solved to find the time.

(5) is just plugging the time from (3) into (4) and simplification of the expression.

3. May 19, 2017

### M Omran

Thanks a million for the crystal clear answer. It is much appreciated.