## Math and physics as code

I've had this intuition for a while now: wouldn't it be neat if you could somehow "copy-and-paste" your physics equations from your textbook and drop them into your computer and that be the code? 

Let's describe a bouncing ball.

Works great to start:

```javascript
const a = (t) => -9.8 /* meters per second sq */ * 10 /* px per meter */
const v0 = 0
const v1 = (t) => v0 + a(t)*t
const p0 = 0
const p1 = (t) => p0 + (((v1(t)+v0)/2)*t)
```

But what about the bounce? There are two problems: 

1) We need to reverse the velocity (multiply it by -1)
2) We need to know at what time the bounce occurs

How about a piece-wise defined function? For starters, let's just hardcode the bounce intervals at 3, and 9:

```javascript
const t_bottom1 = 3 // p1(t) === 0, solve for t
const v2 = (t) => -v1(t_bottom1) + a(t)*(t - t_bottom1)
const p2 = (t) => p1(t_bottom1) + ((v2(t) + v2(t_bottom1))/2)*(t - t_bottom1)

const t_bottom2 = 9 // p2(t) === 0, solve for t
const v3 = (t) => -v2(t_bottom2) + a(t)*(t - t_bottom2)
const p3 = (t) => p2(t_bottom2) + ((v3(t) + v3(t_bottom2))/2)*(t - t_bottom2)

const p = (t) => {
  if (t < t_bottom1) {
    return p1(t)
  } else if (t < t_bottom2) {
    return p2(t)
  } else {
    return p3(t)
  }
}
```
Ok, let's abstract this so the ball can bounce forever-ish:

```javascript
const eq0 = {
  t0: -1,
  v: (t) => 0,
  p: (t) => 0
}
var equations = [eq0]
const intervals = [0].concat(range(3,1000, 6)) // [0, 3, 9, 15, ...]
intervals.forEach((t0, i) => {
  var eq = {}
  
  eq_last = equations[i]
  eq.v_last = eq_last.v(t0)
  eq.p_last = eq_last.p(t0)
  
  eq.t0 = t0
  eq.a = (t) => -9.8 /* meters per second sq */ * 10 /* px per meter */
  eq.v = (t) => -eq.v_last + eq.a(t)*(t - t0) // NEXT: can't decrease by 0.9 because then 6 second hardcoded intervals are fucked up
  // THUS: how do make dynamic
  // MAYBE use something like http://algebra.js.org/ to solve the equations
  // WHICH would eventually lead to allowing mouse click generated invervals
  eq.p = (t) => eq.p_last + (((eq.v(t)+eq.v(t0))/2)*(t - t0))
  
  equations.push(eq)
})

function eqI(t) {
  const eqNext = equations.find(eq => t < eq.t0)
  if (!eqNext) { return equations.length-1 }
  const eqNextIndex = equations.indexOf(eqNext)
  return eqNextIndex - 1
}

const p = (t) => {
  const eq = equations[eqI(t)]
  return eq.p(t)
}
```

Full code to be found [here](http://woofjs.com/create.html#physics-and-math/6).

As said in the code, next steps are:

1) make the intervals dynamic (so we can decrease the speed by 10% each bounce), maybe using something like http://algebra.js.org/ to solve the equations
2) see if we can additionally generate intervals by non-predictable user-input, such as mouse clicks
3) ...turn into flappy bird?

Some notes:

* It was really annoying to code this, because math can be annoying especially inside JS. I wonder how much better it could be in the right kind of environemnt that supports this kind of maths with rapid feedback. I worry that expressing things in this explicit, closed form, with partial equations will be too awkward for people and the gains of mathmatical purity, expresivity won't be worth it. Regardless, I want to follow down this rabbit hole and see where it goes.
* To really show why it's neat to do things in this closed form, I should probably add a time travel debugger thing, that also shows the positions and velocities and accelerations at different points in time, and change when you tweak the params. Probably would be a neat demo.

Author: Steve Krouse