This applet illustrates the concept of Riemann sum, and shows how it converges to the Riemann integral when the number of partitions grows.
The Riemann sum, or numerical integral is shown in green, and the exact integral is shown in red.
Please notice that GeoGebra is not very powerful at integrating, so, for some functions, the exact integral will crash.
Monday, December 24, 2012
Friday, November 23, 2012
Vector field
This applet allows easy visualization of a two dimensional vector field. The equations for both components can be edited. The length of the plotted vectors can also be changed in order to visualize them more neatly.
Vector addition
This applet demonstrates the addition of vectors using the parallelogram rule. The vectors can be edited by dragging the blue dots.
Saturday, November 17, 2012
Relativistic constantly accelerated motion
This applet plots the velocity of a uniformly accelerated body, as seen from an inertial frame of reference. The local acceleration, that is, the acceleration perceived by the travellers, can be edited with the control bar.
As can easily be seen, the velocity approaches asintotically to the speed of light. That is, the acceleration appears to cease as seen from the inertial frame, but its still constant in the moving one.
The time appears to run slower in the moving frame as seen from the inertial one.
Also, the classical results for the same problem can be plotted in order to compare the differences.
As can easily be seen, the velocity approaches asintotically to the speed of light. That is, the acceleration appears to cease as seen from the inertial frame, but its still constant in the moving one.
The time appears to run slower in the moving frame as seen from the inertial one.
Also, the classical results for the same problem can be plotted in order to compare the differences.
Sunday, October 28, 2012
Complex multiplication
This applets illustrates graphically the concept of complex multiplication. As one can easily see, the multiplication of two complex numbers can be performed graphically by multiplying both modules and adding the phase angles.
Saturday, October 27, 2012
Sampling and aliasing
This applet illustrates, in a very simple way, the concept of aliasing. If the sampling frequency is inadequate, the reconstructed signal will be completely erroneous, as can be seen, for example, in the initial setup.
Friday, September 14, 2012
Bézier curves
This applet traces Bézier curves of different orders. The number of control points can be edited, and each control point can be moved by drag and drop.
Friday, September 7, 2012
Dioptric surface
This applet makes a simple ray tracing of a dioptric surface. The source can be moved by drag and drop, and the dioptric surface can be edited. It works fine just for the left surface.
Saturday, September 1, 2012
Mirrors
This applet simulates the behavior of a mirror. The source can be moved by drag and drop, and the mirror's surface can be edited. It works fine for both front and back surfaces.
Sunday, August 19, 2012
Plane curves
This applet allows the user to construct a plane curve using its parametric equations, and display some of its features.
Monday, June 25, 2012
Thin lenses
This applet illustrates the model of thin lenses. Using a negative focal length will provide a divergent lens. The object position can be manipulated dragging the blue dot labeled as object.
Snell's law
This applet illustrates the Snell's law, also known as law of refraction for obvious reasons. The user can drag the blue dot in order to change the incident angle, and manipulate the two refraction indexes.
Thursday, May 24, 2012
First order linear differential equation
This applet plots the slopefield of any linear differential equation, and tries to integrate and plot the solution (given the initial condition).
GeoGebra is not very good at integrating, so it will fail in displaying the exact solution very frequently. The slopefield can always be displayed, even for complicated equations.
GeoGebra is not very good at integrating, so it will fail in displaying the exact solution very frequently. The slopefield can always be displayed, even for complicated equations.
Tuesday, May 22, 2012
Simple molecule
This applet illustrates the simplest model for a ionized hydrogen molecule (two protons + one electron), treated as two hydrogen atoms in the ground state. The user can manipulate the internuclear distance (R).
Questions:
· One of the two states is unable to form stable molecules. Wich one is it?
· What happens to the energy for R >> 0? And for R near to zero?
· Can you estimate the equilibrium distance?
Questions:
· One of the two states is unable to form stable molecules. Wich one is it?
· What happens to the energy for R >> 0? And for R near to zero?
· Can you estimate the equilibrium distance?
Monday, May 21, 2012
Semiconductors
This applet shows the simplest model for a semiconductor crystal. It plots the hole and electron densities, letting the user manipulate all the parameters.
Notice that if the Fermi energy raises, the density of electrons in the conduction band and also the density of holes in the valence band raises too. A practical way to raise the Fermi energy is by adding impurities to the crystal.
Notice that if the Fermi energy raises, the density of electrons in the conduction band and also the density of holes in the valence band raises too. A practical way to raise the Fermi energy is by adding impurities to the crystal.
Fine structure splitting
This applet shows the first energy levels of the hydrogen atom splitted due to the fine structure perturbation. The value of the fine structure constant can be manipulated; using the actual value (1/137) gives an almost imperceptible splitting in the scale of the eV.
Friday, May 18, 2012
Parallel reactions
This applet illustrates the kinetics of a simple parallel reaction (also called competing reaction). The applet plots the relative concentration of each compound against time.
Successive reactions
This applets illustrates the kinetics of a simple two step successive reaction of the form A -> B -> C. The applet plots the relative concentration of each compound against time.
Thursday, May 17, 2012
Reversible reaction
This applets illustrates the evolution of a simple reversible chemical reaction of the form A <--> B. It simply plots the relative concentration of each compound against time.
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