Traffic dynamics problem (model as coupled oscillators, traveling wave)

[anotherghost]

Homework Statement

The problem:
You are given the problem of analyzing the dynamics of a line of cars moving on a one-lane highway. One approach to this problem is to assume that the line of cars behaves like a group of coupled oscillators. How would you set this problem up in a tractable way? Make lots of assumptions.

Homework Equations

The Attempt at a Solution

When people decellerate they do so proportionally to the distance to the car in front of them. People only start breaking when they are less than 4 carlengths away from the next car. (I'm using carlengths as my unit of distance to make things simple.) They only stop when there is 1 carlength between them (as we are measuring from the center of the cars).

Can anyone help me get started on this? I'm having trouble even setting up the differential equation.

 

[mark726]

Thank you for your interesting problem. I would approach this problem by making several assumptions in order to simplify the dynamics of the line of cars and make it more tractable. Here are some assumptions that I would make:

1. All cars in the line have the same mass and size.
2. The cars are moving in a straight line on a one-lane highway.
3. The cars are all moving at a constant speed.
4. The cars are evenly spaced at the beginning of the analysis.
5. The cars follow a linear motion, meaning that they do not change direction or make sudden turns.
6. The cars have a constant deceleration rate when they are less than 4 carlengths away from the car in front of them.
7. The cars come to a complete stop when there is only 1 carlength between them.

With these assumptions in mind, we can set up the problem as a system of coupled oscillators, where each car represents an oscillator. The position of each car can be represented by a variable x, and its velocity by the variable v. We can also define a constant k, which represents the strength of the coupling between the cars.

To simplify the problem, we can also assume that the cars are all moving in the same direction, so we only need to consider the position and velocity in one direction. With these assumptions, we can set up the following differential equation:

m(d^2x/dt^2) = -k(x - 4n) - k(x - 4(n+1)) + k(x - 1(n+1))

In this equation, m is the mass of the car, and n represents the position of the car in the line. The first term on the right side represents the deceleration due to the car in front, the second term represents the deceleration due to the car behind, and the third term represents the deceleration due to the car in front of the car behind.

To solve this differential equation, we can use numerical methods such as Euler's method or Runge-Kutta method. We can also make further assumptions to simplify the problem, such as assuming that the cars are all moving at the same speed, or that the coupling between the cars is constant.

I hope this helps you get started on your problem. Please let me know if you have any further questions or need clarification on any of the assumptions or equations.