Lesson Explainer: Position, Displacement, and Distance | Nagwa Lesson Explainer: Position, Displacement, and Distance | Nagwa

Lesson Explainer: Position, Displacement, and Distance Mathematics • Second Year of Secondary School

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In this explainer, we will learn how to differentiate between position, displacement, and distance traveled, including problems that use vector notation.

The position of a body can be expressed by coordinates in a coordinate system. The coordinate of a position corresponds to a vector 𝑟 from the origin of the coordinate system to the point corresponding to the position of the body, as shown in the following figure.

If the position vector for the body changes from 𝑟 to 𝑟, the displacement of the body from 𝐴 can be represented by the vector 𝑟𝑟=𝐴𝐵=𝑠, as shown in the following figure.

The magnitude of 𝑠 is the length of 𝐴𝐵. The direction of 𝑠 is from 𝐴 to 𝐵.

Let us define displacement in terms of the position of a body at two times, where 𝐴 and 𝐵 represent the positions of the body at the times 𝑡 and 𝑡 respectively.

Definition: Displacement from a Point to another Point

If at a time 𝑡 a body has a position given by the vector 𝑟 and at a later time 𝑡 it has a position given by the vector 𝑟, the displacement from 𝐴 to 𝐵, 𝑠, is given by 𝑠=𝑟𝑟.

It is important to appreciate that the displacement of the body from 𝐴 to 𝐵 does not define the path that the body travels between 𝐴 and 𝐵. Suppose that the body follows the curved dashed path shown in the following figure.

The position vectors for 𝐴 and 𝐵 are the same as they would be if the body had traveled along the line 𝐴𝐵. Any path that the body travels between 𝐴 and 𝐵 produces the same position vectors, and hence the same displacement. The displacement of a body from one point to another is independent of the path taken between the points; only the positions of the start and endpoints of the motion of the body determine the displacement of the body.

Unlike displacement, the distance traveled by a body between two points does depend on the path taken between the points. Let us define distance.

Definition: Distance

Distance is a scalar quantity equal to the length of the path traveled between two points, which is always positive. The shortest distance between two points is the magnitude of the displacement of either point from the other point, which is the straight-line distance between the points, 𝐴𝐵.

While traveling from 𝐴 to 𝐵, a body could travel a greater distance than 𝐴𝐵 by changing its direction of motion as it travels. For example, for a body to travel the curved dashed path shown in the previous figure, the body must change direction throughout its motion.

Consider a body that travels uniformly along a circular path, as shown in the following figure. The body starts its motion at 𝑂 and also ends its motion at 𝑂.

The displacement of the body is zero as it has the same position vector at the start and at the end of its motion. The distance traveled by the body cannot be determined only knowing the position vector for the start point and endpoint of the motion of the body. It is only possible, specifically in the case of a circular path, to determine that the distance traveled must be an integer multiple of the circumference of the circle defining the path, where the integer is equal to the number of times that the body travels the complete length of the circular path.

Let us consider an example demonstrating how distance is determined for a body that changes direction during its motion.

Example 1: Finding the Total Distance Covered by a Person Based on Distance and Direction

A person ran 160 m east and then 175 m north. Find the total distance covered by the person.

Answer

The motion of the person east is a straight-line motion. The distance traveled east is equal to the displacement east, 160 m. The motion of the person north is also a straight-line motion. The distance traveled north is equal to the displacement north, 175 m. The distances traveled by the person are shown in the following figure.

The displacement, 𝑠, is shown in the figure, but this is not what the question requires to be determined. Rather, the distance is required. The distance traveled is equal to the sum of the distances traveled in both of the directions that the person traveled; so, 𝑑, the distance traveled, is given by 𝑑=𝑑+𝑑.eastnorth

Substituting the values given in the question, we find that 𝑑=160+175=335.m

In the previous example, the distance moved was greater than the displacement because the motion was not along a straight line between two points. It is worth noting that even if a body travels from a point 𝐴 to a point 𝐵 only along 𝐴𝐵, it is possible for the body to travel a greater distance than 𝐴𝐵.

This is possible if the body reverses its direction of motion while between 𝐴 and 𝐵. If a body moves along 𝐴𝐵 and, while doing so, at times it travels toward 𝐵 and at other times travels toward 𝐴, the distance traveled by the body in either direction increases the length of the path that the body travels. The body can repeatedly travel along a part of the line in opposite directions, increasing the distance traveled with each repetition.

Let us consider an example demonstrating how displacement and distance are distinguished for a body that moves along a straight line.

Example 2: Finding the Total Distance Traveled and Displacement of a Body Moving in a Straight Line

Using the given figure, calculate the distance 𝑑 and the displacement 𝑠 of a body that moves from point 𝐴 to point 𝐶 then returns to point 𝐵.

Answer

The body travels along a straight-line path throughout its motion, reversing its direction of travel upon reaching the opposite end of the line from which it started.

The motion of the body can be divided into three parts: 𝐴𝐵, 𝐵𝐶, and 𝐶𝐵.

The body travels a distance of 28 cm along 𝐴𝐵, a distance of 24 cm along 𝐵𝐶, and a distance of 24 cm along 𝐶𝐵. Each part of the motion of the body adds to the distance traveled by the body. The distances traveled in both the negative and positive directions are positive values of distance.

The distance traveled, 𝑑, is given by 𝑑=28+24+24=76.cm

The motion of the body in this question is one-dimensional, and so the two possible directions of the displacement are opposite to each other and can be represented as positive or negative values of displacement. We are free to assume that displacement to the right of 𝐴 is positive.

The start point of the travel of the body is 𝐴 and the endpoint is 𝐵. The straight-line distance from 𝐴 to 𝐵 is 28 cm and 𝐵 is to the right of 𝐴, and, therefore, 𝑠=28cm.

The previous example looked at one-dimensional displacement. Now let us see an example where distance and displacement of a body moving in two dimensions are considered.

Example 3: Comparing the Distance Traveled and Displacement of a Body Moving in a Given Path Represented Geometrically

According to the figure, a body moved from 𝐴 to 𝐵 along the line segment 𝐴𝐵, and then it moved to 𝐶 along 𝐵𝐶. Finally, it moved to 𝐷 along 𝐶𝐷 and stopped there. Find the distance covered by the body, 𝑑, and the magnitude of its displacement, 𝑑.

Answer

The distance traveled by the body is the sum of 𝐴𝐵, 𝐵𝐶, and 𝐶𝐷, so 𝑑 is given by 𝑑=6.6+8.8+16.4+12.3=44.1.cm

The magnitude of the displacement of the body is the length of 𝐴𝐷. This can be determined by summing the horizontal and vertical displacements of the body and setting them equal to the opposite and adjacent sides of a right triangle, as shown in the following figure.

The hypotenuse of the triangle has a length equal to 𝑑: 𝑑=(8.8+16.4)+(12.3+6.6)𝑑=(25.2)+(18.9)𝑑=31.5.cm

A body that travels between points takes time to do so, so the position of the body, 𝑟, becomes a function of time, 𝑟(𝑡). Let us look at an example where the displacement of a body in a time interval is considered.

Example 4: Finding the Displacement of a Particle Moving in a Straight Line When Given Its Position as a Function of Time

A particle started moving in a straight line. After 𝑡 seconds, its position relative to a fixed point is given by 𝑟=𝑡4𝑡+7m, 𝑡0. Find the displacement of the particle during the first five seconds.

Answer

This question involves a particle moving along a straight-line path. The displacement of the body along the line can be represented as positive or negative value.

The magnitude of the displacement of the particle at 𝑡=5 is determined by substituting 5 as the value of 𝑡 in the equation 𝑟=𝑡4𝑡+7. Doing this gives us 𝑟(5)=5(4×5)+7𝑟(5)=(25(20)+7)𝑟(5)=12.m

It is important to note, however, that, at 𝑡=0, the value of 𝑟 is given by 𝑟(0)=0(4×0)+7=7,m so between 𝑡=0 and 𝑡=5, the displacement of the particle is 127=5m.

It is important to appreciate that the function 𝑟(𝑡) for the particle is quadratic, with a positive coefficient for 𝑡, and so the displacement of the particle has a minimum value at a time 𝑡min. The value of 𝑡min for a quadratic function of 𝑡, 𝑓(𝑡), given by 𝑓(𝑡)=𝑎𝑡+𝑏𝑡+𝑐, is 𝑡=𝑏2𝑎.min

In this example, 𝑡=(4)2=2.mins

The displacement at 𝑡=2 is 𝑟(2)=2(4×2)+7=3.m

This shows that the particle changes direction after moving 3 metres from its starting point and its displacement at 𝑡=5 is in the opposite direction to the direction of its initial motion.

The position of a body can be represented by a position vector. If the position vector 𝑟 changes with time, then the change in 𝑟 with time in some time interval 𝑡 that is initially zero can be expressed as 𝑠(𝑡)=𝑟(𝑡)𝑟(0).

The difference between the position vector for a body and the displacement for that body in a time interval is demonstrated in the following example.

Example 5: Finding the Displacement Expression of a Body given Its Position Expression Relative to Time

If the position vector of a body at time 𝑡 is given by 𝑟(𝑡)=3𝑡5𝑖+(4𝑡6)𝑗, find its displacement 𝑠(𝑡).

Answer

The question asks for a function of the displacement of the body with time. This can be determined using the formula 𝑠(𝑡)=𝑟(𝑡)𝑟(0), where 𝑟(0)=5𝑖6𝑗.

The change in displacement with time is given by 𝑠(𝑡)=3𝑡5𝑖+(4𝑡6)𝑗5𝑖6𝑗.

This simplifies to 𝑠(𝑡)=3𝑡𝑖+(4𝑡)𝑗.

Only the terms of 𝑟(𝑡) that vary with time are nonzero for 𝑠(𝑡). The 5𝑖 and 6𝑗 terms of 𝑟(𝑡) correspond to the position of the body at 𝑡=0. The position of the body at 𝑡=0 does not affect the change in the position of the body with time.

Key Points

  • The position of a body can be expressed by coordinates in a coordinate system, which correspond to a vector 𝑟 from the origin of the coordinate system to the point corresponding to the position of the body.
  • If a point 𝐴 has a position given by the vector 𝑟 and a point 𝐵 has a position given by the vector 𝑟, the displacement from 𝐴 of 𝐵, 𝑠, is given by 𝑠=𝑟𝑟.
  • The length of the path taken between two points is the distance traveled between those points.
  • If a body changes direction as it travels from its start point to its endpoint, then the distance that it travels is the sum of the distances that it travels in all the directions that it travels in.
  • If a body changes direction as it travels, the distance it travels between two points must be greater than the displacement between the points.
  • The displacement of a moving body can be represented as a function of time by the change in the position vector of the body between instants 0 and 𝑡: 𝑠(𝑡)=𝑟(𝑡)𝑟(0).

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