Simulate a falling object, step by step

The question

Computers don't solve physics equations the way you do on paper. They take lots of tiny steps. Can a few lines of code reproduce what I know about falling objects — and then go further, to things that are hard to work out by hand?

Learn just enough first

Remind yourself of the equations of motion for constant acceleration, especially $s = \tfrac{1}{2}gt^2$
Find a way to run Python: the free Google Colab works in a browser, or install Python with numpy and matplotlib

Kit

A computer with Python (or Google Colab)

My prediction

Without air, how long should a ball take to fall 20 m? Work it out on paper first using $t = \sqrt{2s/g}$.

Method

Type and run this. Each loop moves time forward by a tiny step dt: speed changes by acceleration × dt, and position by speed × dt.

g = 9.81        # m/s², downwards
dt = 0.001      # time step, in seconds
height, speed, t = 20.0, 0.0, 0.0

while height > 0:
    speed = speed + g * dt
    height = height - speed * dt
    t = t + dt

print(f"Hit the ground after {t:.3f} s at {speed:.1f} m/s")

Compare with your paper answer. Then try dt = 0.1 and dt = 0.00001. What happens to the error?
Now add air resistance. The drag force grows with the square of speed: $F = \tfrac{1}{2}\rho C_d A v^2$. Replace the speed line with:

rho, Cd, A, m = 1.2, 0.47, 0.0038, 0.058   # air density, drag coefficient, area and mass of a tennis ball
speed = speed + (g - 0.5 * rho * Cd * A * speed**2 / m) * dt

Drop it from 200 m instead and record the speed at the end. Collect t and speed in lists and plot them with matplotlib.

Data

Time step `dt` (s)	Simulated fall time (s)	Paper answer (s)	Difference
0.1
0.001
0.00001

Result

The speed with drag levels off at the terminal velocity, where drag balances weight:

$$ v_t = \sqrt{\frac{2mg}{\rho C_d A}} $$

Check your simulation against this formula (about 23 m/s for the tennis ball).

Explain it to a friend

A simulation doesn't need clever maths — just "speed changes by a little, position changes by a little", repeated thousands of times. Smaller steps are more accurate. Once that works, you can add things like air resistance that make the paper maths much harder.

New questions

What's the terminal velocity of a ping-pong ball, or a person?
The orbit simulator uses the same stepping idea in two dimensions — try it next.