The trajectory is the geometric line or path described by a moving body. In this section, we are going to study:

- What the position equation is and its relationship with trajectory
- The main types of trajectory equations that exist

## Concept of trajectory and position equation

When a body moves from one point to another, it does so by describing a **geometric line** in space. That geometric line is called **trajectory**, and it is formed by the successive positions of the end of the position vector over time. *Therefore, we often find the coordinates x, y and z of the position vector written as a function of time*, like x(t),y(t) and z(t) to represent the evolution of the position of the bodies with time.

The **trajectory** of a body is the **geometric line** described by a body in motion.

The **position equation** or **trajectory equation** represents the position vector as a function of time. Its expression, in Cartesian coordinates and in three dimensions, is given by:

Where:

: is the$\overrightarrow{r}\left(t\right)$ **position equation**or the**trajectory equation***x(t)*,*y(t),**z(t)*: are the**coordinates**as a function of time :are the$\overrightarrow{i}\text{,}\overrightarrow{j}\text{,}\overrightarrow{k}$ **unit vectors**in the directions of OX, OY and OZ axes respectively

For those problems where you are working in fewer dimensions, you can simplify the previous formula by eliminating unnecessary terms. This way, the position equation:

- In two dimensions becomes
, since$\overrightarrow{r}\left(t\right)=x\left(t\right)\overrightarrow{i}+y\left(t\right)\overrightarrow{j}+\overline{)z\left(t\right)\overrightarrow{k}}=x\left(t\right)\overrightarrow{i}+y\left(t\right)\overrightarrow{j}$ *z=0* - In two dimensions becomes
, since$\overrightarrow{r}\left(t\right)=x\left(t\right)\overrightarrow{i}+\overline{)y\left(t\right)\overrightarrow{j}}+\overline{)z\left(t\right)\overrightarrow{k}}=x\left(t\right)\overrightarrow{i}$ *y=0*and*z=0*

The following animation illustrates the concept of the **position equation** or **trajectory equation**.

### Types of trajectory equation

In addition to the above expression, there are other ways of expressing the trajectory of the motion of a body. Below, we show other **types of position or trajectory equations**:

**Parametric trajectory equations**: Each of the coordinates is established as a function of time in the form x=x(t),y=y(t),z=z(t). For example, the parametric coordinates of a body that moves in the plane x-y could be:- x=t+2
- y=t
^{2}

**Explicit trajectory equation**: It is obtained by removing the parameter t of the previous expressions and solving one variable in function of the other. In our example, it would be:$x=t+2\Rightarrow t=x-2$ $y={t}^{2}\Rightarrow \mathit{y}\mathbf{=}\mathbf{(}\mathit{x}\mathbf{-}\mathbf{2}{\mathbf{)}}^{\mathbf{2}}$

**Implicit trajectory equation**: It is obtained by making*f(x,y)=0*.$(x-2{)}^{2}-y=0$

Take the following example, imagine that a train is moving east 50 meters every second. After the first second the train is located 50 meters from the origin. After second 2, the train is located 100 m from the origin and so on. Therefore, we could write:

- The motion
**x coordinate**as a function of time: m$x=50t$ - The
**position equation**: m$\overrightarrow{r}=50t\overrightarrow{i}$ - The
**distance to the origin**, given by the magnitude of the position vector:$\left|\overrightarrow{r}\right|=50t$