Three charged particles, two positive (blue) and one negative (red) are released from rest at the vertices of an isosceles triangle (equilateral in the initial setting).
It’s assumed that the particles have the same charge (but for the sign), the same mass (inertia) and that only the electric force acts on them.
The system dynamics will be just driven by Coulombian attractive/repulsive forces.
Anyway, given the strong symmetries in the initial conditions and given the conservation of energy and momentum, the system can be reduced to just a couple of differential equations, since the position/velocity of one of the blue particles is enough to set the positions/velocities of the other two. Here’s a video of the resulting “Electric dance”:
On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of . It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and a false alarm rate estimated to be less than 1 event per 203 000 years, equivalent to a significance greater than 5.1σ. The source lies at a luminosity distance of Mpc corresponding to a redshift . In the source frame, the initial black hole masses are M⊙ and M⊙, and the final black hole mass is M⊙, with M⊙ radiated in gravitational waves. All uncertainties define 90% credible intervals. These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.
A new interactive simulation created with the software Wolfram Mathematica, reproducing an anticipatory model for pedestrian interactions, is now available at this page of the CDF simulation section.
Here’s the demo video posted on youtube:
The model is based on the paper:
A universal power law governing pedestrian interactions by Ioannis Karamouzas, Brian Skinner, and Stephen J. Guy
published on 2 December 2014 in Physical Review Letters
The article, together with some other interesting related material, is also available in this page of the Applied Motion Lab, University of Minnesota.
The main point of the model is that the interaction force between two pedestrian is not based on their distance (as it’d happen for, say, electrons) but rather on their time-to-collision, which is defined as “the duration of time for which two pedestrians could continue walking at their current velocities before colliding”.
So, in this model you won’t see nearby pedestrians repel each other if their trajectories are not such to produce a collision in the next few seconds. That makes possible for pedestrians to walk side by side as it happens in the real world.
On the other hand two pedestrians about to collide will try to change their motion (in velocity direction and/or speed).
See the CDF simulationspage for further details.
Too much long to be posted in in the twitter @wolframtap (Wolfram Tweet-a-Program). But short enough to show how some basic mathematical ideas can be very simple and yet beautiful (even if, maybe, useless). Here’s the video posted on youtube:
It’s not easy to understand the mathematical inner workings of moving waves.
That’s because a wave is a perturbation in a media that changes with space and time.
So, even in the simplest case (i.e. waves propagating in a 1-dimensional media like a string) there are at least three variables involved: amount of the perturbation, position and time.
To understand better the relations between those variables and the wave parameters I’ve created, with the software Wolfram Mathematica, an interactive demonstration about moving waves. The demonstration has been exported in the .cdf format so that it can be interactively used with the free CDF Player (see here how to install it).
• The moving wave
• The time view at a fixed position ()
• The space view at fixed times (when the flash comes)
• The 3D view, in which there is also the moving point representing the state of the perturbation at as time goes by and its space-time trajectory.
In the demonstration it will be possible to change the wave parameters and see how its dynamic evolution changes accordingly.
For those who don’t have the free CDF Player installed on the PC (or those visiting this site from a a smartphone/tablet iOS/Android) here is a short video preview of the demonstration:
The anisotropies of the Cosmic microwave background (CMB) as observed by Planck. The CMB is a snapshot of the oldest light in our Universe, imprinted on the sky when the Universe was just 380 000 years old. It shows tiny temperature fluctuations that correspond to regions of slightly different densities, representing the seeds of all future structure: the stars and galaxies of today. (Credits: ESA and the Planck Collaboration)
More info… (link to the ESA’s Planck project website)