Let's explain this of course of how we get the answer. In theory, there will be no absolute answer for this one. We only know the answer using floating-point approximation to returns the smallest interval between two representable numbers by Math.abs(answer - expect) <= 1e-2. I think we can also use Number.EPSILON.

Some may get different result, it does not matter if you get 10 metres with one method and 12.5 metres with another – both are obviously closest to 10, which is thus the right answer.

Calculate for Stopping Distance:

First we calculate the reaction distance by using this equation:

Actually, this is classic high school physics, but let's see the equation

Second we calculate the breaking distance using the reaction distance:

Now both distances are combined to get stopping distance:

Calculate for Average Speed:

Remember that we get the average speed by using equation below:

The question is what is distance and what is time. We can only reverse engineer the equation we did from earlier by getting the velocity from reaction distance equation.

First we will decompose the equation $d = rd+bd$ to $rd = d-bd$ to $bd = d-rd$ :

What will matter here is that we get the velocity from the reaction distance

Now, let's convert this into code:

Calculation for stopping distance:

/* First we calculate the reaction distance: */
    let rd = (v * 1) / 3.6
/* Then we calculate the braking distance: */
    let bd = rd * rd / (2 * mu * 9.81)
/* Now both distances are combined: */
    let d = bd + rd

Calculation for average speed:

let g = 9.81;
/* First decompose stopping distance bd = rd - 1 to rd = bd - 1 */
let d_ = (-(2 * mu * g) + Math.sqrt((2 * mu * g) * (2 * mu * g) - (4 * -d * (2 * mu * g)))) / 2
/* Then average speed by rd = v / 3.6 to  v = rd x 3.6 */
let s_ = d_ * 3.6

Execution Experiment:

And, here's for Sheldon Cooper (☕):

First part of the challenge is almost written out as is, but the second part requires solving quadratic equation.

Solution in Haskell:

dist :: Double -> Double -> Double
dist vKmh mu = d_reaction + d_braking
  where
    d_reaction = v*t_reaction
    d_braking = (v*v) / (2*mu*g)
    v = fromKmh vKmh

speed :: Double -> Double -> Double
speed d mu = toKmh v
  where
    -- Solve using quadratic formula
    v = (-b + (sqrt $ b*b - 4*a*c)) / (2*a)
    a = 1 / (2*mu*g)
    b = 1 / t_reaction
    c = -d

fromKmh :: Double -> Double
fromKmh v = v * 1000 / 3600

toKmh :: Double -> Double
toKmh v = v * 3600 / 1000

g :: Double
g = 9.81

t_reaction :: Double
t_reaction = 1

Tests:

> dist 144 0.3
311.83146449201496

> dist 92 0.5
92.12909477605366

> speed 159 0.8
153.79671564846308

> speed 153 0.7
142.14404997566152

Is there something about this problem I'm missing? This seems like it's just a math problem, is there some trick I'm missing that makes it faster? Or not lose precision? Or am I missing an edge case some how?

anyways, here's a javascript solution.

const g = 9.81; // gravity in m/s
const k = 3600 / 1000; // conversion coefficient: m/s to km/hr

const dist = (v, mu, t=1) => {
  v = v / k; 
  return v * t + v * v / (mu * 2 * g); 
}

const speed = (d, mu, t=1) => {
  return k * (Math.sqrt(g * mu * (2 * d + g * t * t * mu)) - g * t * mu);
}

console.log(dist(144, 0.3)) // 311.83146449201496
console.log(dist(92, 0.5)) // 92.12909477605365
console.log(speed(159, 0.8)) // 153.79671564846308
console.log(speed(153, 0.7)) // 142.14404997566152