averageVelocity :: Time -> Time -> PositionFunction -> Velocity
averageVelocity t0 t1 x = (x t1 - x t0) / (t1 - t0)

averageVelocity2 :: Time -> TimeInterval -> PositionFunction
                 -> Velocity
averageVelocity2 t dt x = (x (t + dt/2) - x (t - dt/2)) / dt

type R = Double

type Time         = R
type TimeInterval = R
type Position     = R
type Velocity     = R

type PositionFunction = Time -> Position
type VelocityFunction = Time -> Velocity

type Derivative = (R -> R) -> R -> R

derivative :: R -> Derivative
derivative dt x t = (x (t + dt/2) - x (t - dt/2)) / dt

carPosition :: Time -> Position
carPosition t = cos t

carVelocity :: Time -> Velocity
carVelocity = derivative 0.01 carPosition

carVelocity' :: Time -> Velocity
carVelocity' t = derivative 0.01 carPosition t

carVelocityAnalytic :: Time -> Velocity
carVelocityAnalytic t = -sin t

velFromPos :: R                   -- dt
           -> (Time -> Position)  -- position function
           -> (Time -> Velocity)  -- velocity function
velFromPos dt x = derivative dt x

positionCV :: Position -> Velocity -> Time -> Position
positionCV x0 v0 t = v0 * t + x0

type Acceleration = R

accFromVel :: R                       -- dt
           -> (Time -> Velocity)      -- velocity function
           -> (Time -> Acceleration)  -- acceleration function
accFromVel = derivative

velocityCA :: Velocity -> Acceleration -> Time -> Velocity
velocityCA v0 a0 t = a0 * t + v0

positionCA :: Position -> Velocity -> Acceleration
           -> Time -> Position
positionCA x0 v0 a0 t = a0 * t**2 / 2 + v0 * t + x0

pos1 :: Time -> Position
pos1 t = if t < 0
         then 0
         else 5 * t**2

vel1Analytic :: Time -> Velocity
vel1Analytic t = undefined

acc1Analytic :: Time -> Acceleration
acc1Analytic t = undefined

vel1Numerical :: Time -> Velocity
vel1Numerical t = undefined

acc1Numerical :: Time -> Acceleration
acc1Numerical t = undefined