The ancient Greeks watched this closely, and from it estimated the distance to the Moon within 10%.
A system of cycles and epicycles accounted well for the observed motions of Venus and Mercury.
A system of cycles and epicycles accounted well for the observed motions of Mars and Jupiter.
A refinement to Ptolemy’s epicycles to account for a small linear oscillation, but made from circular motions.
Demonstrating why observing the path of Mars through the Heavens cannot settle the argument.
Relative orbital sizes and periods of the inner planets: Mercury, Venus, Earth and Mars.
Relative orbital sizes and periods of the outer planets: Jupiter, Saturn, Uranus and Neptune.
Kepler theorized that the planets moved on spherical surfaces separated by having Platonic solids just fit between neighboring orbits: see his model here.
Set a planet in motion, and watch Kepler's Third Law in action.
This applet shows Venus' orbit from different perspectives to understand how the sunlit half of the planet exhibits phasess like the Moon.
Animating Galileo’s own diagram combining steady horizontal motion and constant vertical acceleration to give a parabolic trajectory.
A very accurate applet, plotting multiple trajectories, and including realistic air resistance.
Newton’s realization that the parabolic motion of a cannonball and the circular motion of the Moon were different aspects of the same thing, that the Heavens and the Earth obeyed the same laws, was the key scientific contribution to the Enlightenment.
He began with motion round a regular polygon, with impulses at the corners to change direction, then increasing side numbers to approach a circle. This is essentially how calculus evolved.
Try finding a successful launch strategy: what speed and at what orbital time to land on Mars?
What if the attraction were not inverse square? (It isn’t in General Relativity). And, when are orbits stable?
Once it was clear that light is a wave, what was waving? Light could move through a vacuum (from the Sun and stars). It was assumed that a material called aether must fill all space, and waves as light passed through. This experiment was designed to detect it.
A clock based on bouncing light between two mirrors shows why the constancy of light speed means all moving clocks run slow.
Simple 2D elastic collisions, viewed in the lab frame and the CM frame: a relativistic version makes inertial mass increase with speed understandable.
Rutherford scattered heavy energetic charged particles (“alphas”) from gold atoms to try to find how the positive charge was spread through the atom—the accepted Thomson AtomThomson model had it fairly evenly distributed. To his amazement, he discovered the observed scattering could only be explained by the presence of a very tiny nucleus containing essentially all the mass and charge.
Press Start to see the grid of nuclei, then press for one nucleus to emit a neutron. If that hits another nucleus, two new neutrons come out. There is almost certainly runaway reaction for a big enough grid: above the “critical size”.
Simple harmonic motion is the shadow on a diameter of steady circular motion.
The wave equation follows from F = ma for a tiny bit of string.
A traveling sound wave carries energy, but this applet shows there is no net movement of the air, despite appearances.
Inverted for a fixed end, not for a free end: this can be understood by watching two pulses collide.
Here a change in string density, but the same result as light going from air to glass, for instance.
Observe the change in frequency of sound (or light!) from a moving source.
Synchronized waves from two slits have constructive/destructive interference at points with relative slit distances same phase/opposite phase.
Checking the relative phase from the slits at each point on a screen to construct the interference pattern.
How do they relate? Is one always bigger? Find the answer with this simple applet.
The results can be surprising!
The temperature of a gas is proportional to the kinetic energy of the molecules. Watch how compression heats things up!
This animation correlates the basic Carnot engine cycle to the corresponding P,V graph plus a thermometer.
This applet shows how molecular velocities cause the Brownian Motion of a dust particle. Through the microscope, a tiny particle jitters. With (difficult to achieve!) further magnification we see it's because energetic molecules are pummeling it.
A box has red particles in the left hand half, green in the right. They all begin with the same speed but random direction. Watch as they mix! How long will it take? The vertical lines give the average positions of red and green particles.
Try different numbers of particles and different sizes
Beginning with all molecular speeds equal, the dynamics rapidly evolves to the theoretical speed distribution predicted by Maxwell (with the predicted size-dependent fluctuations).
Adding gravity to colliding molecules dynamics leads to exponential density drop off with height, but uniform temperature.
A dynamic realization: initially equal speeds, and constant gravity along the pipe. Bouncing off the walls, they lose their along the pipe velocity component. The velocity along the pipe approaches a parabolic profile.
The only three-dimensional flat faced solids with every face identical, and every vertex identical.
The path traced by a point on a wheel as the wheel rolls on a flat surface.
The path traced by a point on a wheel as the wheel rolls inside a circle.
A system with two time scales: discover the physics with this applet.
Is this the ideal damping for good shocks?
Plotting together the responses of the oscillator to different initial values, the curves can look completely different, but their difference is simple.
Investigate how a damped simple harmonic oscillator responds to being externally driven at different freequencies: in particular, near resonance.
The phase lag is the key to the power transfer rate. Here you can investigate how it varies with driving frequency and damping.
See the motion of a particle in a simple harmonic oscillator potential with adjustable cubic and quartic terms added.
Near resonance, gradually increasing the driving frequency increases the amplitude—until there is a discontinuous drop!
Explore how gradually increasing the driving force leads to a cascade of period doublings, then to chaos.
In nonchaotic parameter regimes, small initial differences die away; in chaotic regimes they explode. Explore the transition.
In chaos, initially close trajectories diverge exponentially, the Lyapunov exponent quantifies this. Here you can set initial conditions and actually measure the exponent.
Plotting the motion in 2D (position and velocity), gives new insight into period doubling, stability and chaos.
Taking snapshots of the 2D motion one cycle apart gives a few dots in the non-chaotic regime (after initial transients), but for chaos an amazing fractal pattern can emerge—a strange attractor. You can watch it develop with this applet.