HoHCI-BikeSim/Assets/Scripts/BicyclePhysics.cs

using System;
using UnityEngine;

public class BicyclePhysics
{
    private const float K_A = 0.5f; //wind resistance coefficient
    private const float A = 0.6f; //frontal area bike + rider; m2
    private const float D = 1.226f; //air density; kg/m3

    /// <summary>
    /// Returns a speed in m/s for a bike riding a gradient with the same power it would be ridden in the flat with the input speed.
    /// Using formula from http://www.sportsci.org/jour/9804/dps.html. Assuming v=s and neglecting rolling resistance
    /// The speed is calculated by solving k_aAs_1^3d = k_aAs_2^3d + giMs_2 for s_2
    /// </summary>
    /// <param name="speedFlat">Speed the bike would have in flat terrain</param>
    /// <param name="systemMass">Mass of bike + rider</param>
    /// <param name="gradient">gradient in vertical meters gained per meter travelled forward</param>
    /// <returns></returns>
    public static float SpeedAtGradientForSpeedAtFlat(float speedFlat, float systemMass, float gradient)
    {
        var g = -Physics.gravity.y;
        const float k = K_A * A * D;
        var b = g * gradient * systemMass;

        float result;

        if (gradient < -0.01f) // 1 percent
        {
            result = 2 * Mathf.Sqrt(-b / (3 * k)) *
                     Mathf.Cos(
                         Mathf.Acos(3 * Mathf.Sqrt(-3 * k / b) * k * speedFlat
                                    / (2 * b))
                         / 3);
        }
        else
        {
            var sqrtTerm = Mathf.Sqrt(
                Mathf.Pow(b, 3f)
                /
                (27f * Mathf.Pow(k, 3f))
                +
                Mathf.Pow(speedFlat, 6f)
                /
                4f
            );
            var secondTerm = Mathf.Pow(speedFlat, 3f) / 2f;
            result = Qbrt(sqrtTerm + secondTerm) + Qbrt(-sqrtTerm + secondTerm);
        }

        if (float.IsNaN(result))
        {
            Debug.LogWarning(
                $"BicyclePhysics: result is NaN for k = {k} and b = {b} (gradient = {gradient}, mass = {systemMass})");
            return speedFlat;
        }

        return result;
    }

    private static float Qbrt(float x) => x < 0 ? -Mathf.Pow(-x, 1f / 3f) : Mathf.Pow(x, 1f / 3f);
}