 Distance and displacement might seem like similar terms but in physics, understanding the difference can mean getting a question right instead of wrong.

In this lesson, we will define these terms and illustrate how easy it is to confuse the two.

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## 5K: How Far Did I Travel?

Last Thanksgiving, I decided to run my first 5K. I won’t embarrass myself by telling you my exact time, but I was able to successfully finish the race. If you asked the other racers how far I traveled, they would all tell you 5 kilometers. If you asked a group of physicists how far I traveled, they might all tell you 0 kilometers – that I didn’t change position at all.

It might surprise you to learn that both answers are correct. But how can that be? It all depends on the subtle yet very important difference between distance and displacement.

## Distance Versus Displacement

Let’s start by defining these two terms, starting with the easier of the two.Distance is how much ground is covered by an object, regardless of its starting or ending position. There is no directional component to a distance measurement, making it a scalar quantity. So, during my 5K, I ran 5 kilometers total. It doesn’t matter where the starting line or finish line were or in which direction I was running.

It only matters that, if you trace and measure my path, I covered a distance of 5 km of ground. So, when we asked the other runners how far I had traveled, they all answered correctly with a distance measurement of 5 kilometers.Let’s take a minute to look at our other term: displacement, which is an object’s change in position considering its starting position and final position. A displacement measurement does not take into account what route the object took to change position, only where it started and where it ended. It is easiest to picture displacement by locating where the object started and drawing a straight arrow from this point to the point where the object stopped moving. Remember, in physics this arrow is called a vector.

Its length corresponds to the magnitude, or size, of the movement, and the arrow points in the direction of travel. This makes displacement a vector quantity because it incorporates both movement, magnitude and direction.In the shorthand of physics, displacement is written as Δs. ‘Delta’ is a Greek letter shaped like a triangle and it’s used to represent ‘change in.’ The ‘s’ stands for spatial location. So Δs stands for a change in spatial location. You should get comfortable using this shorthand just in case you see questions asking for you to solve for Δs that don’t actually ask for displacement by name.

## Calculating Displacement

Now, before we look at my displacement during my 5K, I want to try and illustrate the concept of displacement. If I walk 3 meters west, then turn and walk 4 meters north, you can easily calculate the distance traveled: 3 m + 4 m = 7 meters. Since this is a distance, there is no need to worry about the direction of travel. Now, let’s determine my displacement. I can draw an arrow that starts where I started walking and extends to the point where I stopped walking. It should look like this, forming a right triangle.

To determine my displacement, all I need to do is determine how long the arrow is using the lengths of the sides and the Pythagorean Theorem.a^2 + b^2 = c^2Since c^2 is your displacement squared, it should be written as (Δs)^2.Δs = √{(3*3) + (4*4)}Δs = √(9 + 16)Δs = 5 meters northwestDoing the math yields a value of 5 meters.

But this is not the displacement. Remember, we need a direction as well. In this case, I ended up northwest of where I started, so my displacement is 5 meters northwest. Notice that I traveled a total distance of 7 m but have only a displacement of 5 meters northwest.Now back to my 5K. You might already be able to figure out what’s going on here.

How can I travel a distance of 5 km but have a displacement of 0? The trick here is that the race route was a big circle; the starting line and finish line were at the same place. So if we draw out the race route and track my progress around the circle, you can see that I did run 5 kilometers around the race route. But try to draw a vector directly connecting my starting position and my ending position without taking into account my route. You only have one point, so you can’t draw a vector. So, my displacement was 0.

## Lesson Summary

Let’s quickly review distance and displacement.Distance is how much ground is covered by an object regardless of its starting or ending position.

In order to determine distance traveled, you must measure the length of the entire path taken by the object. If you run a 5K on a circular course, your distance traveled is 5 kilometers, regardless of where you started and finished. Distance is a scalar quantity.

Displacement is an object’s change in position, only measuring from its starting position to the final position. Displacement only takes into account your position change from the start to the end, not the exact path you took to get there. To calculate displacement, simply draw a vector from your starting point to your final position and solve for the length of this line. If your starting and ending position are the same, like your circular 5K route, then your displacement is 0. In physics, displacement is represented by Δs.

## Learning Outcomes

By completing this lesson, you might strengthen your capacity to:

• Compare distance and displacement, and distinguish between the two
• Recall the meaning of Δs
• Calculate displacement
Posted on / Categories Math 