Education- Geogebra applets

A Lissajous figure, a=2, b=4, c=1.38. Looks familiar? Seen something like this recently?

I have taught AP Physics C at a local high school, and have tutored that and BC Calculus during the last several years. When not actively involved with students I sometimes develop "applets" that can help illustrate various math and physics principles, using Geogebra.

This is a widely-used educational programming "language" that is free and very useful for developing interactive applets. As the name suggests it is sort of a combination of geometry and algebra. It is especially powerful for studies in geometry. Geogebra is highly visual, and coding in it is a bit unusual compared to other languages. But it does have the great advantage of making the coding of interactive apps easy (once you get comfortable with its peculiarities).

There are many hundreds, if not thousands, of applets online that have been contributed by educators worldwide. Go to and click on “Classroom Resources” to search. You can run these applets online in your browser, or download and install Geogebra itself; then download and run the applets locally. You could then develop your own applets, and upload them if you'd like. My applets, along with some PDFs on various physics and math topics, are accessible by going to

Here are a few examples (screen images, these don't execute):

Gravity Tunnel

Geogebra Gravity Tunnel applet

Suppose a tunnel a foot or so in diameter was drilled all the way through the Earth, through the center, and out the other side. Ignoring friction, air resistance, etc, what would happen if you held a baseball right at the tunnel opening and then let go? Some possibilities: (1) the ball wouldn't move; (2) the ball would drop to the center of the Earth and stop and stay there; (3) the ball would drop all the way through the Earth and stop at the other opening and stay there; (4) the ball would go all the way through and keep going out the other side.

To answer, you need to know this fact: the gravitational force inside a uniform-density sphere (the Earth is neither, but we'll assume it is) varies linearly with the distance from the center. Newton proved this, in his Principia. The force is always directed toward the center, it is zero at the center and it is max at the surface (you may remember that outside this sphere gravity follows an "inverse-square" law).

For many people, what the ball will actually do is pretty surprising. Run the applet at Geogebra and watch what happens (place the cursor at the lower left corner, a run button should appear). Adjust the "alpha-2" slider to change the orientation of the tunnel so that it doesn't pass through the center. Try to guess what will happen first, then run it. There are PDFs attached to the applet page with further technical details.

Incidentally, speaking of the world-wide use of Geogebra, I received an email a few years ago asking about this Gravity Tunnel applet; the email was from a student in France.

The Cart and the Ball

Suppose a cart has a spring mechanism such that a ball is launched with it vertically from the cart (i.e., perpendicular to the floor of the cart) at the same instant the cart, initially at rest, starts sliding down a frictionless inclined plane (no wheels). Will the ball fall back down into the cart? Run the applet and see (place cursor at bottom left, the run button should appear). You can vary the tilt angle and the ball's launch velocity. What would happen if there was friction?

Lissajous patterns

I have 2D and 3D versions of these. The idea is that the x-coordinate and y-coordinate that define a figure are each determined by a separate sine function. By changing the relation between the frequencies of these two functions some interesting figures will be drawn. What's really cool is to sweep a phase angle in one of the sine functions; this has the effect of making the figure move. There should be a little button at the lower left to start it moving (put the cursor down there, the button should show up). This sort of display was often used in 50s sci-fi movies, shown on an oscilloscope. Vary the "a" and "b" sliders to see different patterns.

Combining Sine Waves

If two sine waves are close in frequency, and are added together, we get some interesting effects. This phenomenon of "beats" is important for tuning a guitar. When the strings are nearly in tune, you can improve the tuning by playing harmonics on adjacent strings. You can hear a slight wavering in the sound until the strings are in tune. This won't get the tuning correct in an absolute sense, like being in tune with a piano or organ, but the strings will be in tune relative to each other when the beat frequency vanishes.

You can see movement in the applet if you click one of the slider buttons at the bottom (not shown in pic above), then hit the spacebar. When done, spacebar again, then click somewhere on the plot other than a slider, to de-select the one you did, then do the same process for another. The curves will move around in various ways.

Projectile Motion

Projectile motion, with and without drag; Galileo complementary trajectory in blue.

This applet shows a projectile launched at an adjustable angle, both with and without air resistance ("drag"). The x-velocity, y- velocity, and total velocity vectors are shown as the projectile moves through its trajectory. On my Geogebra page there is a separate section with a collection of PDFs with a lot of mathematical detail about projectile motion. These can be downloaded.

Various parameters can be adjusted; do that first, then to run the applet, click on the time slider and press the spacebar.

The blue path at the bottom is the Galileo complementary trajectory. Galileo showed that, if you launch an object with a given initial velocity at a given angle, launching at the same velocity at the complement to that angle will have the same range. Think of throwing a baseball from deep center field to second base. You could lob it at a high angle, or throw it with the same force directly at the base. Of course the lower trajectory will get there sooner, but they both will get to second. (This can be proved easily, today, with the range equation and simple trig identities.)




Retired nuclear scientist. Extensive experience in nuclear science/engineering, from mid-1960s. Peer-reviewed publications. Power reactor control room operator.

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Retired nuclear scientist. Extensive experience in nuclear science/engineering, from mid-1960s. Peer-reviewed publications. Power reactor control room operator.

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