Unprecedented Footage From Costa Rica Shows Tiny Tropical Lizard (Water Anole) “breathing” From An

Tesseract

Beautiful proofs (#4) - When Gauss was a young child...

The legend goes something like this:

Gauss’s teacher wanted to occupy his students by making them add large sets of numbers and told everyone in class to find the sum of 1+2+3+ …. + 100.

And Gauss, who was a young child (age ~ 10) quickly found the sum by just pairing up numbers:

Using this ingenious method used by Gauss allows us to write a generic formula for the sum of first n positive integers as follows:

I love statues like this. How does someone have so much talent?

If unit vectors always scared you for some reason, this neat little trick  from The story of i by Paul Nahin involving complex numbers is bound to be a solace.

It allows you find the tangential and radial components of acceleration through simple differentiation. How about that! 

Have a good one!

** r = r(t),  θ =  θ(t)

Eh?

Seeing the world from under ! 🐠🐠🐠💦🌊 | benthouard

Location: French Polynesia

'Giant Saucer Stones' Landscape Feature, Boulsworth Hill, Briercliffe, Lancashire.

Orbit change of artificial satellite

In order to raise the altitude of an artificial satellite, it is necessary to accelerate it. When the orbital speed increases, the centrifugal force generated by the rotation of the satellite becomes stronger than the gravity of the earth applied to it, then the altitude rises. At this time, a certain amount of energy is returned from the satellite to the vacuum space to reduce the energy debt. Hence the orbital speed decreases and finally drops below the initial speed. The sum of the reduced amount of the momentum energy and the energy required for the acceleration is returned to the vacuum space.

In order to lower the altitude of an artificial satellite, it is necessary to decelerate it. When the orbital speed decreases, the centrifugal force becomes weaker than the gravity, then the altitude drops. At this time, the satellite is given a certain amount of energy from the vacuum space. Hence the orbital speed increases and finally rises above the initial speed. The sum of the increased amount of the momentum energy and the energy released by the deceleration is borrowed from the vacuum space.

Potential energy is energy debt borrowed from vacuum energy bank

Potential energy is defined as the energy difference between the energy of an object at a current position and the energy of the object at a reference position (generally, infinitely distant position) in a force field.

The gravitational potential energy of a combination of an object with mass M and another object with mass m separated by distance R is expressed by the following formula using the gravitational constant G.

The potential energy is zero when the distance between objects is infinite, and the negative energy increases as the distance decreases by gravity.

Because there can be no negative energy in the real space, potential energy should be considered as fictitious energy. Regarding potential energy as “energy debt” is easy to understand. Then what do objects borrow energy from? The answer is the vacuum space. Potential energy is the energy debt borrowed from “the vacuum energy bank”. An increase in the negative energy means an increase in the energy given by the vacuum space.

Therefore, the law of conservation of energy is established only when the vacuum energy is counted.

When an object is attracted to another object by gravity, a certain amount of energy is given to it from the vacuum space. Hence its energy debt increases by the amount given from the vacuum space, and its momentum energy increases then it accelerates.

On the contrary, when applying a force to an object and moving it against gravity, it returns a certain amount of energy to the vacuum space. Hence the energy debt decreases by the amount returned to the vacuum space, and its momentum energy decreases then it decelerates from the initial speed.

We say “potential energy increases” when the energy debt decreases, but we should say “potential energy decreases”.