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

## 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:

$\stackrel{\to }{r}\left(t\right)=x\left(t\right)\stackrel{\to }{i}+y\left(t\right)\stackrel{\to }{j}+z\left(t\right)\stackrel{\to }{k}$

Where:

• $\stackrel{\to }{r}\left(t\right)$ : is the position equation or the trajectory equation
• x(t), y(t), z(t): are the coordinates as a function of time
• $\stackrel{\to }{i}\text{,}\stackrel{\to }{j}\text{,}\stackrel{\to }{k}$ :are the 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 z=0
• In two dimensions becomes , since y=0 and z=0

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

Experiment and Learn

Data

$\stackrel{\to }{r}=$
$\left|\stackrel{\to }{r}\right|=$
Trajectory

The graph shows the trajectory (grey line) that a body represented by a red dot follows over time. In our example, this trajectory is given by the trajectory equation r⃗(t) = (t+1)· i⃗ + (0.05 · t2 + 0.1 · t + 0.05)· j⃗ m, which define at each time t, which is the position vector of the body.

Drag the time slider and check how the body and its position vector change as you move it. The position vector at each time t is obtained by substituting the value of t that you have chosen in the 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:
• $y={t}^{2}⇒\mathbit{y}\mathbf{=}\mathbf{\left(}\mathbit{x}\mathbf{-}\mathbf{2}{\mathbf{\right)}}^{\mathbf{2}}$
• Implicit trajectory equation: It is obtained by making f(x,y)=0.
• $\left(x-2{\right)}^{2}-y=0$

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: $\stackrel{\to }{r}=50t\stackrel{\to }{i}$ m
• The distance to the origin, given by the magnitude of the position vector: $\left|\stackrel{\to }{r}\right|=50t$

## Solved exercises worksheet

Here you can test what you have learned in this section.

#### Position Equation of the S.H.S. Madrid-Barcelona (Spain High Speed train)

difficulty

The S.H.S. Madrid-Barcelona during the first stage of its trip travels east at a rate of 50 m each second and North at 30 m each second. Calculate:

1. The motion x-coordinate as function of time
2. The motion y-coordinate as a function of time
3. The position equation of the train for the studied stretch
4. The distance to the origin of the section as a function of time

## Formulas worksheet

Here is a full list of formulas for the section Trajectory and Equation of Position. By understanding each equation, you will be able to solve any problem that you may encounter at this level.

Click on the icon   to export them to any compatible external program.

#### Position equation in two dimensional Cartesian

$\stackrel{\to }{r}\left(t\right)=x\left(t\right)\stackrel{\to }{i}+y\left(t\right)\stackrel{\to }{j}$

#### Position equation in three dimensional Cartesian

$\stackrel{\to }{r}\left(t\right)=x\left(t\right)\stackrel{\to }{i}+y\left(t\right)\stackrel{\to }{j}+z\left(t\right)\stackrel{\to }{k}$