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# Distance and displacement review

Review key concepts, equations, and common misconceptions related to distance and displacement.

## Key terms

TermMeaning
Coordinate systemSystem we define to describe position. The origin is the point we label as zero. Coordinate systems also define directions for positive and negative numbers.
PositionLocation of an object relative to the origin. We often use the symbol x to refer to position.
DisplacementChange in position of an object. We use the symbol delta, x for displacement, where delta means "change." A vector quantity with units of start text, d, i, s, t, a, n, c, e, end text.
DistanceTotal amount the object has moved. This depends on the whole path traveled, not just the starting and ending points. Distance traveled is always a non-negative number. A scalar quantity with units of start text, d, i, s, t, a, n, c, e, end text.
Reference frameA point of view from which measurements can be made. All frames of reference are equally valid.

## Equations

EquationSymbol breakdownMeaning in words
delta, x, equals, x, minus, x, start subscript, 0, end subscriptdelta, x is the displacement, x is the final position, and x, start subscript, 0, end subscript is the initial positionDisplacement is the difference between the final and initial positions

## Common mistakes and misconceptions

• People sometimes swap the initial and final positions in the displacement equation. It's easy to get mixed up and put the initial position first in the displacement equation to get minus, delta, x instead of delta, x. When calculating displacement, make sure to start with the final position and then subtract the initial position.
• People sometimes think that distance and displacement are just different names for the same quantity. However, distance and displacement are different concepts. If an object changes direction in its journey, the total distance traveled will be greater than the displacement between those two points.
The difference between distance and displacement. Image from Wikimedia Commons, CC BY-SA 3.0
• People sometimes get confused about choosing the “correct” coordinate system. All coordinate systems are equally valid, so we can choose any system. The choice will affect the numbers in the calculation, but as long as we use the coordinate system consistently, the physical meaning of the answer won't change.

For deeper explanations, see our videos:
To check your understanding and work toward mastering these concepts, check out the exercise on finding distance and displacement from graphs.

## Want to join the conversation?

• what is the difference between distance time graphs and position time graphs?
• The gradient of a distance-time graph represents the speed of an object. The velocity of an object is its speed in a particular direction.Note that a motion described as a changing, positive velocity results in a line of changing and positive slope when plotted as a position-time graph.
• How can an Armadillo start at 6m at 0s? What is the question actually meaning? Because when i measure an animal running or moving. i pick 0m and 0s to be right where the animal is currently standing. I just dont randomly decide to say that the origin of my graph is, lets say, 6m behind it haha
• I'd posit that an armadillo doesn't care where the origin of your graph is - you have a nice field laid out in a grid pattern with 0 as the center of the field, but this armadillo with a devil-may-care attitude and a roguish charm just rolls all up in your field, and moves ONLY in one axis because it's apparently 2D cartoon armadillo ;) You get over your shock at seeing a live action cartoon armadillo and finally start measuring its movement, which is at 0 seconds, and the armadillo does whatever armadillo wants, you're just observing at this point.

... hypothetical fictional animals behave very oddly scientific rather than organic, especially in K-12...
• Can someone Please simplify this I still do not understand
• Distance is the length of the path taken by an object whereas displacement is the simply the distance between where the object started and where it ended up. For example, lets say you drive a car. You drive it 5 miles east and then 3 miles west. The distance travelled is 8 miles, but the displacement is 2 miles because you end up 2 miles from where you started. Hope this helps!
• Δx is a vector quantity. Shouldn't the x have a arrow above it?
• It can have the arrow above it, but if it doesn't have one it is still generally understood to be a vector.
• When solving for direction and displacement on a position-time graph, how do you know when the values for each will have the same absolute value (or in other words, despite the displacement possibly being negative)?
• It depends whether or not the direction changes. For the position-time graph you can think of change in direction as crossing the x-axis. If direction does not change then distance and displacement will be the same (distance will not have a sign in front of the value). If the direction does change then the two will vary.
• What is reference Frame?
• Even if the object might be moving vertically, can I still the one-dimensional number line to figure out the displacement or the distance, because it worked for me on some of the practice problems.