Lesson 5 Displacement vs. Distance
Distance, Displacement, and Position
When describing the distance that an object travelled, it is often important to use the point where it started, as a reference.
C5.4 Map of Alberta
The distance an object travelled is a
scalar quantity—it is the length of the path between two points. If you are talking about the position the object ends up compared to where it started, then you are referring to
displacement.
Displacement describes the straight-path distance from one point to another, including its direction.
The symbol for distance is "Δd." The symbol "Δ" is the Greek letter delta, which means change. So Δd means change in the distance of an object from one point to a second point.
The symbol for displacement is "«math»«mo»§#8710;«/mo»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«/math»." Displacement is a vector quantity, so the arrow is written above it.
Let’s look at a practical example of the difference between distance and displacement.
Displacement describes the straight-path distance from one point to another, including its direction.
The symbol for distance is "Δd." The symbol "Δ" is the Greek letter delta, which means change. So Δd means change in the distance of an object from one point to a second point.
The symbol for displacement is "«math»«mo»§#8710;«/mo»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«/math»." Displacement is a vector quantity, so the arrow is written above it.
Let’s look at a practical example of the difference between distance and displacement.
A hockey fan lives in Medicine Hat, Alberta, and wants to travel to Edmonton to watch an NHL hockey game. He checks out the fastest route on a road map.
C5.5 Map of Alberta with route from Medicine Hat to Edmonton
The distance along the highway from Medicine Hat to Edmonton is 560 km.
Δd = 560 km
The blue line in image C5.5 represents this distance.
If a person is able to follow a direct path from Medicine Hat to Edmonton, the distance is 435 km [NW].
«math»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»435«/mn»«mo»§#160;«/mo»«mi»km«/mi»«mo»§#160;«/mo»«mo»[«/mo»«mi»NW«/mi»«mo»]«/mo»«mo»§#160;«/mo»«/math»
The black line in image C5.5 represents this path. It represents displacement, as it describes the straight-path distance from one city to the next, including its direction.
Δd = 560 km
The blue line in image C5.5 represents this distance.
If a person is able to follow a direct path from Medicine Hat to Edmonton, the distance is 435 km [NW].
«math»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mn»435«/mn»«mo»§#160;«/mo»«mi»km«/mi»«mo»§#160;«/mo»«mo»[«/mo»«mi»NW«/mi»«mo»]«/mo»«mo»§#160;«/mo»«/math»
The black line in image C5.5 represents this path. It represents displacement, as it describes the straight-path distance from one city to the next, including its direction.
Calculating Distance and Displacement
When calculating the distance travelled by an object, if the object travels along different paths, you can simply add the distances travelled in each path.
When the direction an object is travelling is indicated, there are standards that indicate if the value is positive or negative.
- [up] is postive.
- [down] is negative.
- [right] is positive.
- [left] is negative.
- [North] is postive.
- [South] is negative.
- [East] is positive.
- [West] is negative.
For example, if a football player back pedals 2.1 m [N], catches the ball, and then runs 7.4 m [S], what distance did the player travel?
To calculate the distance, you would perform the calculation Δd = 21 m + 7.4 m = 9.5 m.
When calculating the displacement of an object, if the object has travelled in more than one path, you need to take into account the direction of each path when adding each length.
In the example of the football player, the player travels 2.1 m [N] and then runs 7.4 m [S]. This is in the opposite direction. So, this needs to be taken into account in the calculation. Also, remember that north is a positive direction and south is a negative direction.
C5.6 Path of football player
To calculate the displacement, you would add the two values, taking into account that north is a positive direction and south is a negative direction.
«math»«mo»§#160;«/mo»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mo»(«/mo»«mo»+«/mo»«mn»2«/mn»«mo».«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»)«/mo»«mo»+«/mo»«mo»(«/mo»«mo»§#8211;«/mo»«mn»7«/mn»«mo».«/mo»«mn»4«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»)«/mo»«mo»=«/mo»«mn»5«/mn»«mo».«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo».«/mo»«mo»§#160;«/mo»«/math»
A negative answer here means that the player has ended up south of his starting position; south is negative.
Or, you could also look at the calculation for displacement as the operation:
«math»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mn»2«/mn»«mo».«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mo»[«/mo»«mi mathvariant=¨normal¨»N«/mi»«mo»]«/mo»«mo»-«/mo»«mn»7«/mn»«mo».«/mo»«mn»4«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mo»[«/mo»«mi mathvariant=¨normal¨»S«/mi»«mo»]«/mo»«mo»=«/mo»«mn»5«/mn»«mo».«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mo»[«/mo»«mi mathvariant=¨normal¨»S«/mi»«mo»]«/mo»«mo»§#160;«/mo»«/math»
Notice that you still find the answer to be that the football player ends up 5.3 m south of his original position.
«math»«mo»§#160;«/mo»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mo»(«/mo»«mo»+«/mo»«mn»2«/mn»«mo».«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»)«/mo»«mo»+«/mo»«mo»(«/mo»«mo»§#8211;«/mo»«mn»7«/mn»«mo».«/mo»«mn»4«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»)«/mo»«mo»=«/mo»«mn»5«/mn»«mo».«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo».«/mo»«mo»§#160;«/mo»«/math»
A negative answer here means that the player has ended up south of his starting position; south is negative.
Or, you could also look at the calculation for displacement as the operation:
«math»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mn»2«/mn»«mo».«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mo»[«/mo»«mi mathvariant=¨normal¨»N«/mi»«mo»]«/mo»«mo»-«/mo»«mn»7«/mn»«mo».«/mo»«mn»4«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mo»[«/mo»«mi mathvariant=¨normal¨»S«/mi»«mo»]«/mo»«mo»=«/mo»«mn»5«/mn»«mo».«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mo»[«/mo»«mi mathvariant=¨normal¨»S«/mi»«mo»]«/mo»«mo»§#160;«/mo»«/math»
Notice that you still find the answer to be that the football player ends up 5.3 m south of his original position.
Examples:
- A stray cat is out at night and walks eastward 5.4 m, and then turns and walks westward 8.9 m.
- What distance does the cat travel?
To calculate the distance, you would perform the calculation Δd = 5.4 m + 8.9 m = 14.3 m. - What is the displacement (magnitude and direction) of the cat?
To calculate the displacement, you would add the two values, taking into account that east is a positive direction and west is a negative direction.
«math»«mo»§#160;«/mo»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mo»(«/mo»«mo»+«/mo»«mn»5«/mn»«mo».«/mo»«mn»4«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»)«/mo»«mo»+«/mo»«mo»(«/mo»«mo»§#8211;«/mo»«mn»8«/mn»«mo».«/mo»«mn»9«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»)«/mo»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#8211;«/mo»«mn»3«/mn»«mo».«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«/math»
Because west is negative, then an answer of –3.5 m means that you could express it as «math»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mo»§#8211;«/mo»«mn»3«/mn»«mo».«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mn»3«/mn»«mo».«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mo»[«/mo»«mi mathvariant=¨normal¨»W«/mi»«mo»]«/mo»«mo».«/mo»«mo»§#160;«/mo»«/math»
- A child runs across the playground 12.3 m [N] and then 16.8 m back to the south.
- What distance does the child travel?
To calculate the distance, you would perform the calculation Δd = 12.3 m + 16.8 m = 29.1 m.
-
What is the displacement (magnitude and direction) of the child?
To calculate the displacement, you would add the two values, taking into account that north is a positive direction and south is a negative direction.
«math»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mo»(«/mo»«mo»+«/mo»«mn»12«/mn»«mo».«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»)«/mo»«mo»+«/mo»«mo»(«/mo»«mo»§#8211;«/mo»«mn»16«/mn»«mo».«/mo»«mn»8«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»)«/mo»«mo»=«/mo»«mo»§#8211;«/mo»«mn»4«/mn»«mo».«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo».«/mo»«mo»§#160;«/mo»«/math»
Because south is negative, then an answer of –4.5 m means that you could express it as «math»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mo»§#8211;«/mo»«mn»4«/mn»«mo».«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mn»4«/mn»«mo».«/mo»«mn»5«/mn»«mo»§#160;«/mo»«mo»[«/mo»«mi mathvariant=¨normal¨»S«/mi»«mo»]«/mo»«mo».«/mo»«/math»
- A kayaker paddles 32 m [N], then 6 m [S], and then 16 m [N] again.
-
What distance does the kayaker travel?
To calculate the distance, you would perform the calculation Δd = 32 m + 6 m + 16 m = 54 m.
- What is the displacement (magnitude and direction) of the kayaker?
To calculate the displacement, you would add the three values, taking into account that north is a positive direction and south is a negative direction.
«math»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mo»(«/mo»«mo»+«/mo»«mn»32«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»)«/mo»«mo»+«/mo»«mo»(«/mo»«mo»§#8211;«/mo»«mn»6«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»)«/mo»«mo»+«/mo»«mo»(«/mo»«mo»+«/mo»«mn»16«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»)«/mo»«mo»=«/mo»«mo»+«/mo»«mn»42«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo».«/mo»«mo»§#160;«/mo»«/math»
Because north is positive, then an answer of +42 m means that you could express it as «math»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mo»+«/mo»«mn»42«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mn»42«/mn»«mo»§#160;«/mo»«mo»[«/mo»«mi mathvariant=¨normal¨»N«/mi»«mo»]«/mo»«mo».«/mo»«mo»§#160;«/mo»«/math»
Do you want a bit more detailed explanation of how to calculate the distance and displacement of an object? Watch this video for more information on how to complete these calculations.
https://adlc.wistia.com/medias/wzrxuitjpd
Read This
Please read pages 137 to 138 in your Science 10 textbook. Make sure you take notes on your readings to study from later. You should focus on how distance and displacement are measured, calculated, and communicated. Remember, if you have any questions, or do not understand something, ask your teacher!Digging Deeper
There are two methods that can be used to measure and communicate an object’s direction. They both use the mathematical method of a coordinate system grid.
Cartesian Method: uses a coordinate system grid with an x-axis and y-axis similar to a graph. Directions are started from the x-axis, which is the starting point at 0˚, and directions are determined in a counterclockwise direction.
C5.8 Navigator Method
Navigator Method: uses a coordinate system grid with the directions of north [N], south [S], east [E], and west [W]. Directions are started from north, which is the starting point at 0˚, and directions are determined in a clockwise direction.
Practice Questions
Complete the following practice questions to check your understanding of the concept you just learned. Make sure you write complete answers to the practice
questions in your notes. After you have checked your answers, make corrections to your responses (where necessary) to study from.- A soccer player runs down the field for 2.14 m [S] and then turns and runs 7.45 m back [N].
- What distance does the player travel?
To calculate the distance, you would perform the calculation
«math»«mo»§#160;«/mo»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mrow»«/mrow»«/mover»«mo»=«/mo»«mn»2«/mn»«mo».«/mo»«mn»14«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#160;«/mo»«mn»7«/mn»«mo».«/mo»«mn»45«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»=«/mo»«mn»9«/mn»«mo».«/mo»«mn»59«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo».«/mo»«mo»§#160;«/mo»«/math» - What is the displacement (magnitude and direction of the player?
- A cross-country skier skies 24 m [W], then 57 m [E], and then 68 m [W] again.
- What distance does the skier travel?
To calculate the distance, you would perform the calculation
«math»«mo»§#160;«/mo»«mi»§#916;«/mi»«mi»d«/mi»«mo»=«/mo»«mn»24«/mn»«mo»§#160;«/mo»«mtext»m«/mtext»«mo»+«/mo»«mn»57«/mn»«mo»§#160;«/mo»«mtext»m«/mtext»«mo»+«/mo»«mn»68«/mn»«mo»§#160;«/mo»«mtext»m=149§#160;m«/mtext»«mo».«/mo»«mo»§#160;«/mo»«/math» - What is the displacement (magnitude and direction) of the skier?
To calculate the displacement, you would perform the calculation
«math»«mo»§#160;«/mo»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mn»24«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»-«/mo»«mn»57«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»+«/mo»«mn»68«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»=«/mo»«mn»35«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mn»35«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mo»[«/mo»«mi mathvariant=¨normal¨»W«/mi»«mo»]«/mo»«mo».«/mo»«mo»§#160;«/mo»«/math»
To calculate the displacement, you would perform the calculation
«math»«mo»§#160;«/mo»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mn»2«/mn»«mo».«/mo»«mn»14«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»-«/mo»«mn»7«/mn»«mo».«/mo»«mn»45«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»=«/mo»«mo»§#8208;«/mo»«mn»5«/mn»«mo».«/mo»«mn»31«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mn»5«/mn»«mo».«/mo»«mn»31«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mo»[«/mo»«mi mathvariant=¨normal¨»N«/mi»«mo»]«/mo»«mo».«/mo»«mo»§#160;«/mo»«/math»
«math»«mo»§#160;«/mo»«mi»§#916;«/mi»«mover»«mi»d«/mi»«mo»§#8594;«/mo»«/mover»«mo»=«/mo»«mn»2«/mn»«mo».«/mo»«mn»14«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»-«/mo»«mn»7«/mn»«mo».«/mo»«mn»45«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»=«/mo»«mo»§#8208;«/mo»«mn»5«/mn»«mo».«/mo»«mn»31«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mn»5«/mn»«mo».«/mo»«mn»31«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»§#160;«/mo»«mo»[«/mo»«mi mathvariant=¨normal¨»N«/mi»«mo»]«/mo»«mo».«/mo»«mo»§#160;«/mo»«/math»
Watch This
Distance and Displacement ©ADLC https://adlc.wistia.com/medias/9ubslvsli4
This video will provides you with a great wrap-up of the difference between distance and displacement, and this section of Lesson 4.