What is the Difference between Trajectory and Displacement?

The Main difference between trajectory and displacement Is that the latter is the distance and direction traveled by an object, while the first is the path or form 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.

Graph showing the displacement, trajectory and distance.

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, being a vectorial magnitude, measures of length known as centimeters, meters or kilometers are used.

The formula for calculating the displacement is defined as follows:

What is the Difference between Trajectory and Displacement?

From which it follows that:

  • Δ X = Displacement
  • X F = Final position of the object
  • X I = Initial position of the object

Example of displacement

1- If a group of children are at the beginning of a course, 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 replacing 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 from X F , Minus 50m which is the value of X I , Gives us as a result 70m, that is, when reaching the 120m routes the displacement was 70m to the right.

4- We proceed to solve similarly 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 is to say 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, generally adopts geometric forms like the straight line, parabola, circle or ellipse. It is identified through an imaginary line and being a scalar quantity is measured in meters.

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

The equation for calculating the trajectory of an object in the International System is given by:

What is the Difference between Trajectory and Displacement?  1

Of which we have to:

  • R (t) = is the equation of the trajectory
  • 2t - 2 y t 2 = Represent the coordinates as a function of time
  • . I y . J = are the unit vectors

To understand the calculation of the trajectory traversed by an object we will 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 trajectory equation is a function of X, to do this define the values ​​of X and Y respectively in each of the vectors raised:

1- Solve the first position vector:

  • R (t) = (2t + 7) . I + t 2 . J

2 - Ty = f (x), where X is given by the contents of the unit vector . I and Y is given by the contents of the unit vector . J:

  • X = 2t + 7
  • Y = t 2

3- y = f (x), ie time is not part of the expression so we must clear it, we have:

What is the Difference between Trajectory and Displacement?  2

4- We substitute the Y-clearance. It stays:

What is the Difference between Trajectory and Displacement?  3

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

What is the Difference between Trajectory and Displacement?  4

As we can see, it gave us a second degree equation, which means that the trajectory has the form of a parabola.

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 we saw above y = f (x), we must clear 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- Resolving the parentheses we have the equation of the resulting path for the second unit vector:

What is the Difference between Trajectory and Displacement?  5

In this procedure resulted in a straight line, which tells us that the trajectory has a straight line.

Understanding the concepts of displacement and trajectory 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 with an arrow.
  • It uses measures of length (centimeter, meter, kilometer).
  • It is a vector quantity.
  • Take into account the direction (right or left)
  • 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 displacement is zero.
  • The module must coincide with the space to travel as long as the path is a straight line and no changes occur in the direction to follow.
  • The module tends to increase or decrease as the movement occurs, keeping in mind the path.

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.
  • It depends on a reference system.
  • When the starting point or starting position is the same as the end position, the path is given by the distance traveled.
  • The value of the path coincides with the modulus of the displacement vector, if the resulting path is a straight line, but there is no change in the direction to follow.
  • It always increases when the body moves, regardless of the path.

References

  1. Alvarado, N. (1972). ) Physical. First Year of Science . Editorial Fotoprin C.A. Venezuela.
  2. Fernández, M; Fidalgo, J. (2016) ). Physics and Chemistry 1º Bachillerato. Ediciones Paraninfo, S.A. Spain.
  3. Guatemalan Institute of Radiophonic Education. (2011) Fundamental physics . First Semester Grupo Zaculeu. Guatemala.
  4. Fernández, P. (2014) Scientific and technological field. Editions Paraninfo. S.A. Spain.
  5. Fisica Lab (2015) Vector Displacement. Retrieved from: fisicalab.com.
  6. Examples of (2013) Displacement. Retrieved from: examplesde.com.
  7. Salon Home Project (2014) What is displacement? Recovered from: salonhogar.net.
  8. Physics Lab (2015) Concept of trajectory and position equation. Retrieved from: fisicalab.com.


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