Trajectory and Equation of Position

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.

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 $\overrightarrow{r}\left(t\right)=x\left(t\right)\overrightarrow{i}+y\left(t\right)\overrightarrow{j}+\cancel{z\left(t\right)\overrightarrow{k}} =x\left(t\right)\overrightarrow{i}+y\left(t\right)\overrightarrow{j}$, since z=0
• In two dimensions becomes $\overrightarrow{r}\left(t\right)=x\left(t\right)\overrightarrow{i}+\cancel{y\left(t\right)\overrightarrow{j}}+\cancel{z\left(t\right)\overrightarrow{k}} =x\left(t\right)\overrightarrow{i}$, since 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=t²
• 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: $x=50t$ m
• The position equation: $\overrightarrow{r}=50t\overrightarrow{i}$ m
• The distance to the origin, given by the magnitude of the position vector: $\left|\overrightarrow{r}\right|=50t$