What is the total length of the path Travelled by a moving object and is specified by a magnitude only?

In this explainer, we will learn how to define distance as the length of a path between two positions and displacement as the straight-line distance between two positions.

Let us first discuss distance.

When an object moves from one point to another, it moves along a path that connects those points. The path that the object moves along has a length. This length equals the distance that the object moves.

A path between two points can be a straight line between them. The following figure shows a straight path that an object moves along.

A path between two points can also be curved, as shown in the following figure.

For both straight lines and curves, it makes no difference to the distance traveled which point the object travels from and which it travels to, as the length of the line is the same either way. A distance does not have a direction, only a magnitude.

Quantities that have magnitudes but not directions are scalar quantities; hence, distance is a scalar quantity.

When an object moves, the motion can be between more than two points.

Suppose that an object travels from a point A to a point B, and then from point B to a point C, as shown in the following figure.

The movement of the object can be split into the movement from A to B and the movement from B to C.

The distance, 𝑑, that the object travels is given by 𝑑=()+().distancefromAtoBdistancefromBtoC

Let us look at an example in which the distance traveled along a path that changes direction is determined.

What is the total distance walked by someone along the lines shown in the diagram?

Answer

The person walks along three straight lines. The distance that they move is the sum of the lengths of these lines. The distance moved, 𝑑, is given by 𝑑=15+10+20=45.m

We can see that distance always increases as an object moves. The least distance that an object can move is zero, when it remains at rest.

Let us now look at another example in which the distance traveled along a path that changes direction is determined.

What is the total distance covered by someone who walks along the lines shown in the diagram?

Answer

The person walks along three straight lines. The distance that they move is the sum of the lengths of these lines. The distance moved, 𝑑, is given by 𝑑=5+8+7=20.m

Suppose that the object also travels from point C back to point A, as shown in the following figure.

The distance, 𝑑, that the object travels is now given by 𝑑=()+()+().distancefromAtoBdistancefromBtoCdistancefromCtoA

We suppose that the object repeatedly travels from A to B, from B to C, and from C to A, making the journey several times. We can call the number of times that the object repeats this journey 𝑛.

We can now call the distance that the object travels 𝐷, which is given by 𝐷=𝑛𝑑.

Let us look at an example in which the distance traveled along a closed path is determined.

What is the total distance covered by someone who walks along the lines shown in the diagram, not walking on any line more than once?

Answer

The person walks along three straight lines. No line is walked more than once and no line is not walked, so each line is walked once.

The distance that the person moves is the sum of the lengths of these lines. The distance moved, 𝑑, is given by 𝑑=6+6+6=18.m

What has been shown in these examples for distances moved in straight lines also applies to distances moved along curved paths.

Suppose that an object travels along a circular path, as shown in the following figure.

Let us suppose also that the object travels once around the circular path, returning to its starting point and not reversing direction. The distance that the object moves equals the circumference of the circle.

Suppose instead that an object travels along the path shown in the following figure that takes the object from A to B, then from B to C, and finally from C to A.

The distance that the object moves is the sum of the lengths of the curved paths between the points.

Distance has now been explained.

Let us now discuss displacement.

When an object changes position, as well as moving a distance, it also has a displacement.

Displacement is also a quantity that describes the separation of points from each other, but it is not the same thing as distance.

The reason for displacement being different from distance is that displacement has a direction. Quantities that have a direction as well as a magnitude are vector quantities, so displacement is a vector quantity. Displacement is often represented by the symbol 𝑠.

Consider the line connecting the points shown in the following figure.

An object could move from A to B or from B to A. The displacement of the object traveling from A to B is in the opposite direction to the displacement of the object traveling from B to A.

Let us suppose the distance from A to B is 1 metre. This is the same as the distance from B to A.

The displacement of an object that moves from A to B is 1 metre, but the displacement of an object that moves from B to A is −1metre, as shown in the following figure.

We can see from this that the distance between A and B equals the magnitude of the displacement of an object moving from A to B and it equals the magnitude of the displacement of an object moving from B to A. The direction of the displacement is shown by the positive or negative sign of the displacement.

The direction that is positive is from A to B in this example. Which direction is considered positive can be freely chosen. Whichever direction is considered positive, the opposite direction must be considered negative.

A displacement has a direction, so a displacement between two points must be a straight line between the points. A curved path changes direction along its length, so it does not have one specific direction.

Let us now look at an example in which displacements of points from other points are determined.

A speedboat passes by markers at the points A, B, and C, as shown in the diagram. Positive displacement is considered to be away from A, toward C.

• What is the boat’s displacement from A when it is at B?
• What is the boat’s displacement from C when it is at B?
• What is the boat’s displacement from A when it is at C?
• What is the boat’s displacement from C when it is at A?

Answer

The positive direction for displacement is stated in the question to be from A toward C. This is true whatever point the question asks for the displacement to be taken from.

When the boat is at B, the displacement from A to B is in the same direction as from A toward C, so it is in the positive direction, as shown in the following figure.

The distance from A to B is given by the distance from A to C minus the distance from B to C, so the displacement from A to B is given by 𝑠=250−180=70.m

When the boat is at B, the displacement from C to B is in the opposite direction to the direction from A toward C, so it is in the negative direction, as shown in the following figure.

The distance from C to B is 180 m, so the displacement from C to B is given by 𝑠=−180.m

When the boat is at C, the displacement from A to C is in the positive direction, as shown in the following figure.

The distance from A to C is 250 m, so the displacement from A to C is given by 𝑠=250.m

When the boat is at A, the displacement from C to A is in the negative direction, as shown in the following figure.

The distance from C to A is 250 m, so the displacement from C to B is given by 𝑠=−250.m

An object can return to its starting point by moving some distance along a line and then reversing the same distance along that line. The following figure shows points A and B connected by a straight line.

If an object travels from A to B and back to A, it has zero displacement. The distance moved by the object will not be zero, however, but will be twice the distance from A to B.

Let us now look at an example in which the distance and displacement due to the motion of an object that reverses direction are compared.

A leaf is blown by the wind. The leaf moves 5 m forward and then 3 m backward.

• What is the distance moved by the leaf?
• What is the leaf’s net forward displacement?

Answer

The leaf moves in a straight line forward a distance of 5 m and then moves in a straight line backward a distance of 3 m. The distance that the leaf moves is the sum of the lengths of these paths. The distance moved, 𝑑, is given by 𝑑=5+3=8.m

The question asks for the net forward displacement of the leaf, so we should consider the forward motion of the leaf to be positive and must therefore consider the backward motion of the leaf as negative. The net forward displacement of the leaf is given by 𝑠=5+(−3)=5−3=2.m

If the motion of an object includes a change of direction that is not a complete reversal of that direction, then the object does not move along one line. The object can then be considered as having displacement in the 𝑥-direction and in the 𝑦-direction, as shown in the following figure.

The object travels equal distances in the 𝑥-direction and in the 𝑦-direction. The object has two displacements, each in a different direction.

Let us now look at an example in which the displacements in the 𝑥 and 𝑦 directions of an object that moves are determined.

A person walks from point A to point B, as shown in the diagram.

• What is the displacement of point B from point A in the 𝑥-direction?
• What is the displacement of point B from point A in the 𝑦-direction?

Answer

The diagram shows that the positive 𝑥-direction is to the right. The object moves 4 m to the right and also moves 1 m to the left. The displacement in the 𝑥-direction is given by 𝑠=4+(−1)=4−1=3.m

The diagram shows that the positive 𝑦-direction is upward. The object moves 3 m upward and also moves 5 m downward. The displacement in the 𝑦-direction is given by 𝑠=3+(−5)=3−5=−2.m

An object can return to its starting point by moving along a closed path that changes direction. The path that the object takes to return to its starting position can consist of straight lines, curves, or both straight lines and curves, as shown in the following figure.

In the closed paths shown in the preceding figure, only the straight lines can represent displacements. Only displacements are vectors, so only the straight lines have arrows showing a direction.

Let us look at an example involving the displacements in the 𝑥- and 𝑦-directions of objects that move along closed paths.

Two people walk along triangular lines that connect the points A, B, and C shown in the diagram. The first person walks from point A along a triangular path that returns them to point A. When the first person returns to point A, they stop. The second person walks from point B along a triangular path that returns them to point B. When the second person returns to point B, they stop.

• What is the displacement of the first person from point A in the 𝑥-direction when they stop?
• What is the displacement of the first person from point A in the 𝑦-direction when they stop?
• What is the displacement of the second person from point B in the 𝑥-direction when they stop?
• What is the displacement of the second person from point B in the 𝑦-direction when they stop?

Answer

The first person starts at point A and walks a triangular path back to point A, where they stop. Point A is the point at which the motion of the first person starts and the point at which it ends. The displacement of the person is, therefore, zero. A displacement of zero is zero in any direction, so the displacement in the 𝑥-direction is zero and the displacement in the 𝑦-direction is zero.

The motion of the second person is almost exactly the same as that of the first person, the only difference being that the second person starts at point B rather than point A.

The different starting positions of the two people make no difference to their displacements, as each person returns to their starting position and so both have zero displacement.

Let us now summarize what has been learned in these examples.

• A distance is the length of a path between two points.
• The path between points can be a straight line or a curve.
• The direction that an object moves between two points has no effect on the distance that the object moves. Distance has magnitude but no direction, so it is a scalar quantity.
• The total distance moved by an object that moves between multiple points is the sum of the distances that it moves between those points.
• A displacement is a straight-line distance from one point to another point.
• A displacement has a direction as well as a magnitude, so it is a vector quantity.
• For motion along a line, a direction must be chosen from one end of the line to the other for which the displacement is taken as positive. For the opposite direction, the displacement is taken as negative.
• The magnitude of the displacement along a straight-line path between two points is the distance between those points along that path.
• The motion of an object that travels from a point back to that same point produces zero displacement
• For the motion of an object including changes of direction that are not a complete reversal of direction, the object will have displacements along more than one line.