A reference system is a coordinate system in which elements in space and time can be represented.
Important elements of the coordinate system are defined in the following links.
A reference system is very important in physics, as it enables us to locate an object and determine its state of motion. In addition, the reference system enables us to establish the type of motion observed as a function of the observer's position. The relativity of motion is also dependent on the position of the reference system, since motion can be perceived differently depending on the observer's position.
A player kicks a soccer ball which follows a parabolic trajectory with a maximum height of 10 m and a range of 50 m. The ball's horizontal and vertical displacement can be plotted assuming the observer is seated in the stands. We can plot its horizontal and vertical displacement, assuming the observer is seated in the stands.
Note: Image in English coming soon.
It's not just displacement that can be plotted in a reference system. Speed, acceleration and time are other examples of variables that can be placed in a reference system.
The graphs below show the characteristics (position, speed and acceleration) of a moving object descending an inclined plane as a function of time.
Note: Image in English coming soon.
Note: Image in English coming soon.
Note: Image in English coming soon.
A coordinate system associates a point with precise coordinates that allow it to be located in space.
There are two types of coordinates:
Cartesian coordinates are |(x, y)| coordinates used to locate a point in a Cartesian plane relative to a point of origin.
When you want to locate an object, you can use three axes:
- the x-axis, or abscissa axis, which corresponds to the horizontal axis;
- the y-axis, or ordinate axis, which corresponds to the vertical axis;
- the z axis, used to position an object according to depth (the third dimension).
Generally speaking, in physics problems, we will only use the first two axes.
The Cartesian coordinates of the point |\small \text {A}| shown in the Cartesian plane of origin |\small \text {O}| are |\small (2, 3)|.
Cartesian coordinates often represent measurable data with units of measurement. It is important to consider the context in order to know which units of measurement should be used to locate a particular point.
Polar coordinates are coordinates of the type |\small (r, \theta)|, which allow you to locate a point using the distance between the starting point and the end point, i.e. the radius |\small r|, as well as the measure of the angle with respect to the positive abscissa axis, i.e. the angle |\small \theta|.
To determine the polar coordinate, we need to determine the starting point, i.e. the pole. This starting point is a bit like the origin of a Cartesian plane. Once this point has been established, we determine the value of the radius r by measuring the distance between the starting point and the point we want to locate.
To determine the value of the angle |\small \theta|, we need to determine the value of the angle between the positive x-axis, which is our starting point, and the radius determined in the previous step.
The polar coordinates of point A on the plane below, whose pole is O, are |\small (3.61; 56.31^{\circ})|.
Cartesian coordinates can be converted to polar coordinates using Pythagoras' theorem and trigonometric relations.
To change from Cartesian coordinates |\small (x, y)| to polar coordinates |\small (r, \theta)|, use the following rules:
|| r = \sqrt{x^2 + y^2}|| || \theta= \arctan\left(\frac{y}{x} \right)||
A point B is located at coordinates |(2, 2)|. What are the polar coordinates of point B?||\begin{align}r = \sqrt{x^2 + y^2} \quad \Rightarrow \quad
r &= \sqrt{2^2 + 2^2} \\
&= \sqrt{8} \\
&\approx 2,83 \end{align}||
||\begin{align}\theta=\arctan\left(\frac{y}{x} \right) \quad \Rightarrow \quad
\theta&=\arctan\left(\frac{2}{2} \right) \\
&= \arctan\left(1\right) \\
&= 45^{\circ} \end{align}||
The polar coordinates of point B are therefore |(2.83; 45^{\circ})|.
Polar coordinates can be converted to Cartesian coordinates using trigonometric relations.
To change from polar coordinates |\small (r, \theta)| to Cartesian coordinates |\small (x, y)|, use the following rules:
||x=r\times \cos \theta|| ||y=r\times \sin \theta||
The polar coordinates of a point C are |\small (3, 30^{\circ})|. What are its Cartesian coordinates?
||\begin{align}x=r\times \cos \theta \quad \Rightarrow \quad
x &= 3\times \cos 30^{\circ} \\
& \approx 2,6 \end{align}||
||\begin{align}y=r\times \sin \theta \quad \Rightarrow \quad
y &= 3\times \sin 30^{\circ} \\
&= 1,5 \end{align}||
The Cartesian coordinates of point C are therefore |(2,6; 1,5)|.