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Spring work

When a force is applied on a spring, and the length of the spring changes by a differential amount dx, the work done is Fdx. For linear elastic springs, ... more

Hawking Radiation - Temperature of a black body (or a black hole)

A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. A black hole ... more

Compton scattering

Compton scattering is an inelastic scattering of a photon by a free charged particle, usually an electron. It results in a decrease in energy (increase in ... more

Electron's speed at any radius

The electrons can only orbit stably, without radiating, in certain orbits at a certain discrete set of distances from the nucleus. These orbits are ... more

Hawking radiation energy of black-body (Planck) spectrum

black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. A black hole ... more

Triangulation (surveying)

In surveying, triangulation is the process of determining the location of a point by measuring only angles to it from known points at either end of a fixed ... more

Worksheet 334

In a video game design, a map shows the location of other characters relative to the player, who is situated at the origin, and the direction they are facing. A character currently shows on the map at coordinates (-3, 5). If the player rotates counterclockwise by 20 degrees, then the objects in the map will correspondingly rotate 20 degrees clockwise. Find the new coordinates of the character.

To rotate the position of the character, we can imagine it as a point on a circle, and we will change the angle of the point by 20 degrees. To do so, we first need to find the radius of this circle and the original angle.

Drawing a right triangle inside the circle, we can find the radius using the Pythagorean Theorem:

Pythagorean theorem (right triangle)

To find the angle, we need to decide first if we are going to find the acute angle of the triangle, the reference angle, or if we are going to find the angle measured in standard position. While either approach will work, in this case we will do the latter. By applying the cosine function and using our given information we get

Cosine function
Subtraction

While there are two angles that have this cosine value, the angle of 120.964 degrees is in the second quadrant as desired, so it is the angle we were looking for.

Rotating the point clockwise by 20 degrees, the angle of the point will decrease to 100.964 degrees. We can then evaluate the coordinates of the rotated point

For x axis:

Cosine function

For y axis:

Sine function

The coordinates of the character on the rotated map will be (-1.109, 5.725)

Reference : PreCalculus: An Investigation of Functions,Edition 1.4 © 2014 David Lippman and Melonie Rasmussen
http://www.opentextbookstore.com/precalc/
Creative Commons License : http://creativecommons.org/licenses/by-sa/3.0/us/

Center of mass - Barycentric coordinates

In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to ... more

Speed

In everyday use and in kinematics, the speed of an object is the magnitude of its velocity (the rate of change of its position); it is thus a scalar ... more

Momentum (Time)

Momentum is a measure of an object tendency to move in a straight line with constant speed. Momentum has a direction and can be used to predict the ... more

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