A falling stone: average speeds over shrinking intervals

"""A curved graph: average speeds of a falling stone over shrinking intervals.

In the idealised model of a stone dropped from rest (no air resistance), it
has fallen d(t) = 4.9 t^2 metres after t seconds. Its distance-time graph is
curved, so the average rate of change depends on the interval. The
experiment computes, with exact fractions, the average speed in each of the
first four seconds, then the average speed over intervals [2, 2 + h] for
shrinking h (and for intervals ending at 2), and compares the results with
the exact formula 19.6 + 4.9h proved in the lesson. As h shrinks the average
speeds approach 19.6 m/s: a preview of the instantaneous speed (derivative).
"""

from fractions import Fraction as F

G_HALF = F(49, 10)  # 4.9 m/s^2, half the acceleration due to gravity

def fallen(t):
    """Distance fallen after t seconds in the model d(t) = 4.9 t^2."""
    return G_HALF * F(t) ** 2

def avg_speed(a, b):
    """Average rate of change of d between times a and b (a != b)."""
    if a == b:
        raise ValueError("the two times must differ")
    return (fallen(b) - fallen(a)) / (F(b) - F(a))

def formula(h):
    """The lesson's exact formula for the average speed over [2, 2 + h]."""
    return F(98, 5) + G_HALF * F(h)

def show(x):
    """A fraction with a decimal beside it, e.g. '1029/50 (= 20.58)'."""
    x = F(x)
    if x.denominator == 1:
        return str(x)
    return f"{x} (= {float(x):.4f})"

def main():
    print("1. Second by second (exact values; all are whole tenths)")
    for t in range(1, 5):
        print(f"   after {t} s: fallen {float(fallen(t)):5.1f} m;"
              f" average speed during second {t}:"
              f" {float(avg_speed(t - 1, t)):4.1f} m/s")
    print("   The rate grows each second: a curved, steepening graph.")
    print()
    print("2. Average speed over the interval from t = 2 to t = 2 + h")
    hs = [F(1), F(1, 10), F(1, 100), F(1, 1000)]
    for h in hs + [-h for h in hs]:
        rate = avg_speed(2, 2 + h)
        mark = "matches" if rate == formula(h) else "DIFFERS"
        print(f"   h = {str(h):>7}: {show(rate):<26} {mark} 19.6 + 4.9h")
    print("   As h shrinks (from either side) the values close in on 19.6.")
    print()
    print("Epistemic status: exact arithmetic in an idealised model (no air")
    print("resistance). Each value agrees with the formula 19.6 + 4.9h, which")
    print("the lesson proves by algebra for every h other than 0; that formula,")
    print("not these eight values, shows the averages approach 19.6 m/s.")

if __name__ == "__main__":
    main()