-------------

AP System = Accident Prevention System

While sitting in my class bored, sipping on my energy drink to keep concentration in class, I was thinking of a new AP system for cars.

Last year, I took a whole physics sequence (the hard one for Physics majors vs other majors), if there is on thing I got out of the classical motion class was:

Using the property that the difference of positions in respect to time is velocity and the difference of velocities with respect to time is acceleration, we get this relation (where x represents distance, t represents time):

a = constant acceleration

dx/dt(dx/dt(x)) = a

or

dx/dt(x) = at + c

or

x(t) = a/2t^2 + v_0t + r_0

Obviously I skipped initial conditions to save space, but we get that under the assumption of a constant acceleration, the distance can be mapped out given an initial velocity and initial position.

How does this have anything to do with AP with cars?

Let car 1 position be mapped with x_1(t) and car 2 position be mapped with x_2(t). If we take everything to a unique reference point, a_i, v_0i, r_0i are all defined => x_i(t) can be defined. Let two cars be in an accident, which implies that x_1(t) = x_2(t) + E, for E is a constant defined to be relatively small. Since E is relatively small, we can take a specific case where E is equal to 0.

=> x_1(t) = x_2(t)

=> x_1(t) - x_2(t) = x_2(t) - x_1(t) = 0

What does this mean? It means, if you take the difference of any two positions and subtract them to get into a neighborhood around 0 with this neighborhood having a increasingly small epsilon, this implies that the cars have got into an accident.

However, when we drive, who drives with constant acceleration? A computer program can calculate these formulas hundreds of times per second. What does this mean, as we calculate the formula hundreds of times per second, we will be using the acceleration given about a millisecond and this acceleration between intervals will be pretty much constant not violating the assumption.

Driving isn't a one dimensional motion, why am I using a one dimensional model? Simple, when you are driving, you are usually traveling in a one dimensional direction not counting the road moving up and down as you are driving. The altitude can easily be calculated without the use of a formula and cars that have significantly greater altitude (such as an overpass) won't be detected as a warning.

---

**Conclusion**

Using GPS services,

With today's GPS Technology, it is easy to draw information such as a cars acceleration, position with a unique reference point and initial velocities. Using a position function, we can make inferences whether or not a car is about to get into an accident. If the difference of two objects positions is approaching 0 at a defined rate, chances are these two objects are heading for a collision. The car could warn the user seconds before an impact with some type of notification settings.

Neglecting security concerns that your car is being mapped by a GPS to prevent accidents, this could be developed to be a new AP system for cars.