For this first assignment, your goal is to set up node.js and npm on your system, and to get familiar with the basics of how to use modules. The intention here is to just focus on understanding the basics, and so it is going to be a little more structured than usual. Also unlike in future assignments, this assignment will be broken up into stages.

In the first stage, your goal is to implement an extremely simplified physics simulation. You will be modeling the motion of a collection of non-interacting moving particles in a box. You can assume that there is no friction and no external forces (ie gravity).

To do this, you should make a module which defines a function that integrates the state of the system forward by some time step. You should export this function as a module. Here is a method signature to help get you started

`module { var nextPosition = positionlength var nextVelocity = velocitylength // Compute next state here return position: nextPosition velocity: nextVelocity }`

The arguments for the method are as follows:

`position`

is an array of length 2 arrays of numbers representing the position of each particle`velocity`

is an array of length 2 arrays of numbers representing the velocity of each particle`ground`

is a length 2 array of numbers representing the normal direction for the bottom of the box`dt`

is the amount of time to step the simulation forward

For each particle, you should update the system using the rule that:

`positioni += dt * velocityi`

If a particle hits one of the sides of the box or crosses the ground side, then the position is integrated up to the time of intersection and the velocity is reflected by the normal of the side. The shape of the box is determined by the following system of inequalities:

`-1 < x < 1`

`y < 1`

`ground[0] * x + ground[1] * y > 0`

You can assume that:

- The position of all particles will be within the box
- The second component of the ground normal will always be greater than 0.

In this project you can use any modules on npm that you would like, or if you prefer you can also write and publish your own modules if you think they will help you with this assignment.

For the first part of this assignment, you should submit an archive that has the following things:

- All the code you wrote to make your project work (excluding code within node_modules/)
- A
`package.json`

file that lists all dependencies within your project and has a reference to the entry point to your project. - A collection of test cases contained in the test/ folder

To verify that your modules is working as intended, you should write at least a couple of test cases. To get you started, here is a simple example:

`var assert = var stepSimulator = var position = 0 05var velocity = 0 10var ground = 0 1var dt = 01 forvar t=00; t<10; t+=dt var nstate = position = nstateposition velocity = nstatevelocity assert assert`

You should write at least two different test cases of your own to supplement this.

To submit your assignment, use moodle and upload a zipped archive of all the above files.

To figure out how to reflect the velocities, think back to the first lecture. Recall that if we have a vector and we want to reflect it about a normal n, the formula is:

`v_reflected = v - 2 * n * dot(n, v) / dot(v, v)`

For more information, see the following wiki article:

To figure out when a particle intersects one of the bounding lines, you need to do a bit of algebra. Remember that the position of the particle at time $t$ is given by,

`p(t) = p_0 + t * v_0`

Where p_0 is the initial position at the start of the time step and v_0 is the initial velocity. To figure out the point in time when the particle hits one of the walls, we need to solve for t in the following equation:

`dot(p(t) - d * n, n) = 0`

where d is the distance from the line to the origin and n is the unit normal of the line.