The **difference between trajectory and displacement** is that the latter is the distance and direction traveled by an object, while the trajectory is the route or the form that the movement of that object takes.

However, to see more clearly the differences between displacement and trajectory, it is better to explain through examples that allow a better understanding of both terms.

**Displacement**

It is understood as the distance and direction traveled by an object, taking into account its initial position and its final position, always in a straight line. For its calculation, since it is a vectorial magnitude, length measurements known as centimeters, meters or kilometers are used.

The formula to calculate the displacement is defined as follows:

From which it follows that:

– Δx = offset

– Xf = final position of the object

– Xi = initial position of the object

**Scroll Example**

1. If a group of children are at the beginning of a route, whose initial position is 50 m, moving in a straight line, determine the displacement at each of the points Xf.

– Xf = 120m

– Xf = 90m

– Xf = 60m

– Xf = 40m

2. The data of the problem is extracted by substituting the values of X2 and X1 in the displacement formula:

– Δx = ?

– Xi = 50m

– Δx = Xf – Xi

– Δx = 120m – 50m = 70m

3. In this first approach, we say that Δx is equal to 120 m, which corresponds to the first value of Xf we find, minus 50 m, which is the value of Xi, which gives us 70 m as a result, that is, when we reach 120 m traveled the displacement was 70 m to the right.

4. We proceed to solve in the same way for the values of b, c and d

– Δx = 90m – 50m = 40m

– Δx = 60m – 50m = 10m

– Δx = 40m – 50m = – 10m

In this case, the displacement gave us negative, that means that the final position is in the opposite direction to the initial position.

**Trajectory**

It is the route or line determined by an object during its movement and its valuation in the International System, it generally adopts geometric forms such as a straight line, parabola, circle or ellipse.

It is identified through an imaginary line and because it is a scalar quantity it is measured in meters.

It should be noted that to calculate the trajectory we must know if the body is at rest or in motion, that is, it is subject to the reference system that we select.

The equation to calculate the trajectory of an object in the International System is given by:

Of which we have to:

– r

– 2t – 2 and t2 = represent the coordinates as a function of time

– ** .**iy **.**j = are the unit vectors

To understand the calculation of the trajectory traveled by an object we will develop the following example:

Calculate the equation of the trajectories of the following position vectors:

– r

– r

First step: As a trajectory equation is a function of X, to do so, define the values of X and Y respectively in each of the proposed vectors:

1. Solve for the first position vector:

– r

2. Ty=f(x), where X is given by the content of the unit vector **.**ie Y is given by the content of the unit vector **.**J:

– X = 2t + 7

– Y = t2

3. y=f(x), that is, time is not part of the expression, therefore, we must clear it, we are left with:

4. We replace the clearance in Y. It remains:

5. We solve the content of the parentheses and we have the equation of the resulting trajectory for the first unit vector:

As we can see, it gave us a second degree equation as a result, this means that the trajectory has the shape of a parabola.

Second step: We proceed in the same way to calculate the trajectory of the second unit vector:

1. r

–X = t – 2

– Y = 2t

2. Following the steps that we saw previously y=f(x), we must clear the time because it is not part of the expression, we are left with:

–t = X + 2

3. We replace the clearance in Y, leaving us:

– y = 2 ( X + 2)

4. Solving the parentheses, we are left with the equation of the resulting trajectory for the second unit vector:

This procedure gave us a straight line, which tells us that the trajectory has a rectilinear shape.

Once the concepts of displacement and trajectory are understood, we can deduce the rest of the differences that exist between both terms.

**More differences between displacement and trajectory**

**Displacement**

– It is the distance and direction traveled by an object taking into account its initial position and its final position.

– It always occurs in a straight line.

– It is recognized with an arrow.

– Use measures of length (centimeter, meter, kilometer).

– It is a vector quantity.

– Take into account the direction traveled (to the right or to the left)

– It does not consider the time spent during the tour.

– It does not depend on a reference system.

– When the start point is the same as the start point, the offset is zero.

– The module must coincide with the space to be traveled as long as the trajectory is a straight line and there are no changes in the direction to be followed.

– The module tends to increase or decrease as the movement occurs, taking into account the trajectory.

**Trajectory**

It is the route or line determined by an object during its movement. It adopts geometric shapes (straight, parabolic, circular or elliptical).

– It is represented by an imaginary line.

– It is measured in meters.

– It is a scalar quantity.

– It does not take into account the direction traveled.

– Consider the time spent during the tour.

– Depends on a reference system.

– When the starting point or initial position is the same as the final position, the trajectory is given by the distance traveled.

– The value of the trajectory coincides with the magnitude of the displacement vector, if the resulting trajectory is a straight line, but there are no changes in the direction to follow.

– It always increases when the body moves, regardless of the trajectory.

**References**

Fernandez, M., Fidalgo, J. (2016). Physics and Chemistry 1st Baccalaureate. Paraninfo Editions, SA Spain.

Guatemalan Institute of Radio Education (2011) Fundamental Physics. First Semester Zaculeu Group. Guatemala.