WHAT IS THE DIFFERENCE BETWEEN TRAJECTORY AND DISPLACEMENT? - SCIENCE - 2025

The main difference between trajectory and displacement is that the latter is the distance and direction traveled by an object, while the former is the path or shape that the movement of that object takes.

However, to see more clearly the differences between displacement and trajectory, it is better to specify its conceptualization 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, as it is a vector magnitude, the measurements of length known as centimeters, meters or kilometers are used.

The formula to calculate the displacement is defined as follows:

From which it follows that:

• Δ _x = displacement
• X _f = final position of the object
• X _i = initial position of the object

Displacement example

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

• X _f = 120m
• X _f = 90m
• X _f = 60m
• X _f = 40m

2- The data of the problem are extracted by substituting the values ​​of X ₂ and X ₁ in the displacement formula:

• Δ _x =?
• X _i = 50m
• Δ _x = X _f - X _i
• Δ _x = 120m - 50m = 70m

3- In this first approach we say that Δ _x is equal to 120m, which corresponds to the first value we find of X _f, minus 50m which is the value of X _i, it gives us as a result 70m, that is, when we reach 120m traveled the displacement was 70m to the right.

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

• Δ _x = 90m - 50m = 40m
• Δ _x = 60m - 50m = 10m
• Δ _x = 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 evaluation in the International System, generally adopts geometric shapes such as the 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 movement, that is, it is subjected 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 (t) = is the equation of the path
• 2t - 2 and t ² = represent the coordinates as a function of time
• . iy. j = are the unit vectors

To understand the calculation of the path traveled by an object, we are going to develop the following example:

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

1. r (t) = (2t + 7) . i + t ² . j
2. r (t) = (t - 2) . i + 2t . j

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

1- Solve the first position vector:

• r (t) = (2t + 7) . i + t ² . j

2- Ty = f (x), where X is given by the content of the unit vector . i and Y is given by the content of the unit vector . j:

• X = 2t + 7
• Y = t ²

3- y = f (x), that is, time is not part of the expression therefore we must solve it, we have:

4- We substitute the clearance in Y. It remains:

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

As we can see, it resulted in an equation of the second degree, this means that the trajectory has a parabola shape.

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

r (t) = (t - 2) . i + 2t . j

• 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 have:

• t = X + 2

3- We substitute the clearance in Y, remaining:

• y = 2 (X + 2)

4- Solving the parentheses we have the equation of the resulting trajectory for the second unit vector:

In this procedure the result was 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 happens in a straight line.
• It is recognized by an arrow.
• Use length measurements (centimeter, meter, kilometer).
• It is a vector quantity.
• Take into account the direction traveled (to the right or left)
• It does not consider the time spent during the tour.
• It does not depend on a reference system.
• When the starting point is the same starting point, the offset is zero.
• The module must coincide with the space to travel as long as the path is a straight line and there are no changes in the direction to follow.
• The modulus tends to increase or decrease as movement occurs, keeping in mind the trajectory.

Trajectory

It is the path 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.
• It 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 path coincides with the module of the displacement vector, if the resulting path 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

1. Alvarado, N. (1972) Physics. First Year of Science. Editorial Fotoprin CA Venezuela.
2. Fernández, M; Fidalgo, J. (2016). Physics and Chemistry 1st Baccalaureate. Ediciones Paraninfo, SA Spain.
3. Guatemalan Institute of Radio Education. (2011) Fundamental Physics. Zaculeu Group First Semester. Guatemala.
4. Fernández, P. (2014) Scientific-technological field. Paraninfo editions. SA Spain.
5. Fisica Lab (2015) Vector Displacement. Recovered from: fisicalab.com.
6. Examples Of. (2013) Displacement. Recovered from: examplesde.com.
7. Living Room Home Project (2014) What is displacement? Recovered from: salonhogar.net.
8. Fisica Lab (2015) Concept of trajectory and position equation. Recovered from: fisicalab.com.