# Your Complete Guide to Understanding the Concepts of Distance and Displacement

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Everybody understands the concept of distance. It is a commonly used term that gives a measure of the length or of the interval between two places or points. In the study of motion in physics distance is not enough and a second quantity called Displacement is needed to understand the change in position of objects.

At first glance distance and displacement seem the same but the two quantities are entirely different in what they represent. Before delving into the Distance vs Displacement calculations, applications and differences, it is essential to understand some of the fundamentals of the concepts.

Position and Frame of Reference

The motion of objects is defined using the change in position of the object with reference to another object or a ‘frame of reference’. To describe the position of a say, a vehicle, it is important to specify a frame of reference.

Typically the Earth is considered as a frame of reference for most of the motion studied. The position of the vehicle can then be specified using all the stationary objects on the Earth such as a tree or a building.

The position of objects can also be determined using other moving objects that are in motion relative to the Earth. However, the frame of reference is chosen depending on the problem being solved.

If a person is walking inside a moving bus, the bus will be chosen as the frame of reference. Although the bus is moving relative to the Earth, a person standing still inside the bus will be considered stationary with reference to the bus.

Understanding Displacement

Displacement simply put is the motion of an object relative to a frame of reference. Consider the example of the person standing in the middle of the bus. If he moves from his initial position then it can be said that his position has changed and this change is called displacement.

Using X as the variable to indicate the position horizontally, mathematically displacement can be defined as the change in position ΔX. If X(f) is the final position of the man and X(o) is the initial position, then the displacement is the difference between the final position and the initial position.

ΔX = X(f) – X(o)

ΔX is only half the story of displacement. The person inside the bus can either move forward or to the front of the bus or move back to the rear of the bus. Therefore it is important to indicate the direction of motion of the person.

If the person moves to the front then ΔX is Positive, if the person moves to the rear then ΔX is negative. Therefore while ΔX gives the magnitude of the displacement, the positive or the negative sign indicates the direction of the displacement. Displacement is therefore a vector quantity.

Consider the case where the person inside the bus moves to the front of the bus and comes back to his original position or seat. Now it can be said that his displacement is zero as he is back to the same position he started from.

Understanding Distance

Using the example of the person inside the bus, distance simply put is the magnitude of the displacement of the person. If the person moves to the front of the bus by 2m, then the distance between his first position and final position is 2m.

Since distance is a Scalar quantity, it has no direction and is always positive regardless of whether the person moves to the front or the rear of the bus.

If the person moves from his original position 2m to the front of the bus and comes back to the original position then it can be said that his displacement is zero but the distance travelled is 2m 2m=4m.

What is Distance travelled?

The difference between distance and displacement truly begins to stand out when the distance travelled is being considered. Although the person in the bus has zero total displacements, he did travel some distance during the activity.

Consider a second example where a person needs to drive around a lake from his home to the grocery store. The store is located 2km across the lake from the house; however, the person needs to take a road that goes around the lake for 5km to reach the store.

In this case, although the person’s displacement from his home is only 2km, the distance travelled or ‘distance’ is 5km.

Quick Concepts Refresher

• Displacement refers to the change in position of an object with respect to a frame of reference while distance gives the measure of the interval between the initial and final points.
• Displacement is a vector quantity with both magnitude and direction. Distance is a scalar quantity that has only magnitude.
• Distance and displacement can both be equal in magnitude when the object travels in a straight line from the initial to the final point.
• The magnitude of displacement can only be equal to or less than the magnitude of the distance.
• Displacement is said to be zero if the object returns to its original position, however, distance travelled can never be zero.

Some common mistakes made while working with distance and displacement

• It is easy to mistake the magnitude of displacement with the magnitude of distance. Going back to the lake example, the magnitude of the displacement between the house and the store can only be 2km. However, depending on the road taken, the distance between the house and the store can be 2km, 5, km or even 10km.
• While calculating the displacement, sometimes people make the mistake of mixing up the final and initial position and end up with a negative value of displacement. It is important to always remember to subtract the Initial position from the Final position.
• In the beginning, people can think that distance and displacement are different terms that describe the same quantity. This is especially the case when the magnitudes of both are the same and create confusion.
• The type of coordinate system chosen can affect the outcome of the problem being solved. Therefore it is important to have the frame of reference right before working on the problems.

Conclusion

While distance and displacement may appear to be the same at first glance, they both describe different quantities. With visual examples, the difference can be clearly seen.

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