Project 4: A Simple Physics Engine

Build a 2D physics simulator with gravity, collisions, and basic integration.

Project Overview

Attribute Value
Difficulty Level 2: Intermediate
Time Estimate 2-3 weeks
Main Language Python
Alternative Languages C++, Rust
Knowledge Area Kinematics and calculus
Tools Plotting tool or simple GUI
Main Book “Physics for Scientists and Engineers” by Serway & Jewett

What you’ll build: A simulator that updates positions and velocities over time, with simple collision response.

Why it teaches math: You apply derivatives and integrals to real motion and see errors when integration is wrong.

Core challenges you’ll face:

  • Implementing time integration (Euler, semi-implicit)
  • Handling collision detection and response
  • Keeping the simulation stable

Real World Outcome

You will simulate bouncing balls or blocks under gravity, producing trajectories and collision events.

Example Output:

$ python physics.py --objects 3 --duration 10
Simulation steps: 600
Collisions: 14
Saved trajectory plot to physics.png

Verification steps:

  • Check energy loss or conservation depending on settings
  • Validate that objects fall at expected rates

The Core Question You’re Answering

“How do derivatives and integrals become a time-based simulation?”

This is calculus in motion.

Concepts You Must Understand First

Stop and research these before coding:

  1. Newton’s laws
    • How does force relate to acceleration?
    • Book Reference: “Physics for Scientists and Engineers” by Serway & Jewett, Ch. 4
  2. Numerical integration
    • Why does Euler integration drift over time?
    • Book Reference: “Numerical Methods for Engineers” by Chapra & Canale, Ch. 25
  3. Conservation
    • What does energy conservation tell you about your simulator?
    • Book Reference: “Physics for Scientists and Engineers” by Serway & Jewett, Ch. 7

Questions to Guide Your Design

  1. Integrator choice
    • Will you use explicit Euler or semi-implicit Euler?
    • How will you choose the time step size?
  2. Collision handling
    • Will you use simple elastic collisions or add restitution?
    • How will you prevent objects from sinking into each other?

Thinking Exercise

Euler Step

Given a velocity of 10 m/s and gravity -9.8 m/s^2, compute velocity after a 0.1s step using Euler integration.

Questions while working:

  • How does step size affect accuracy?
  • What happens if you use a much larger step?

The Interview Questions They’ll Ask

Prepare to answer these:

  1. “What is Euler integration and why is it unstable?”
  2. “How do you detect collisions in 2D?”
  3. “What does conservation of energy tell you about your model?”
  4. “Why does time step size matter?”
  5. “How would you improve simulation accuracy?”

Hints in Layers

Hint 1: Starting Point Start with a single object under gravity.

Hint 2: Next Level Add collision against the floor with a restitution coefficient.

Hint 3: Technical Details Use semi-implicit Euler for better stability.

Hint 4: Tools/Debugging Plot position vs time to see if motion matches expectation.

Books That Will Help

Topic Book Chapter
Newton’s laws “Physics for Scientists and Engineers” by Serway & Jewett Ch. 4
Numerical integration “Numerical Methods for Engineers” by Chapra & Canale Ch. 25
Energy conservation “Physics for Scientists and Engineers” by Serway & Jewett Ch. 7

Implementation Hints

  • Keep physics state in clear structures (position, velocity).
  • Start with simple circles to simplify collisions.
  • Add friction only after basic collisions work.

Learning Milestones

  1. First milestone: You can simulate motion under gravity.
  2. Second milestone: You can detect and resolve simple collisions.
  3. Final milestone: You can explain stability and integration errors.