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Coefficient of restitution ( two objects)

The coefficient of restitution (COR) of two colliding objects is typically a positive real number between 0.0 and 1.0 ... 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/

Barycenter (Two-body problem)

barycentre; from the Greek βαρύ-ς heavy + κέντρ-ον centre) is the center of mass of two or more bodies that are orbiting each other, or the point around ... more

Half-Life

Half-life is the amount of time required for the amount of something to fall to half its initial value. The term is very commonly used in nuclear physics ... more

Amagat's law

Amagat’s law or the Law of Partial Volumes of 1880 describes the behaviour and properties of mixtures of ideal (as well as some cases of non-ideal) ... more

Tsiolkovsky rocket equation as function of payload

The Tsiolkovsky rocket equation, classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that ... more

Rayleigh number (for geophysical applications - related to bottom heating of the mantle from the core)

In fluid mechanics, the Rayleigh number (Ra) for a fluid is a dimensionless number associated with buoyancy-driven flow, also known as free convection or ... more

Time-varying instantaneous voltages for L1 (Three-phase electric application)

In electrical engineering, three-phase electric power systems have at least three conductors carrying alternating current voltages that are offset in time ... more

Time-varying instantaneous voltages for L3 (Three-phase electric application)

In electrical engineering, three-phase electric power systems have at least three conductors carrying alternating current voltages that are offset in time ... more

Kardashev's civilization rating

The Kardashev’s civilization rating is sugested by Carl Sagan from Kardashev scale.

The Kardashev scale is a method of measuring a ... more

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