L1 Distance, Displacement, and Velocity - Part 3
Completion requirements
Unit 7B
Integrals Part 2
Lesson 1: Distance, Displacement, and Velocity
There is a practical application for both the average rate of change and the instantaneous rate of change.
Recall, the instantaneous velocity from part b in the video can also be calculated using derivatives instead of first principles, as follows.
In science courses, or more specifically physics courses, the following formulas are usually introduced.
Note distance and displacement are equivalent when an object travels in a straight line without any changes in direction. Therefore, speed and velocity are equivalent under the same conditions.
Recall, the instantaneous velocity from part b in the video can also be calculated using derivatives instead of first principles, as follows.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»16«/mn»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»t«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»32«/mn»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mn»5«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»32«/mn»«mfenced»«mn»5«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»160«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»ft«/mi»«mo»/«/mo»«mi»sec«/mi»«/mtd»«/mtr»«/mtable»«/math»
In science courses, or more specifically physics courses, the following formulas are usually introduced.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»speed«/mi»«mo»=«/mo»«mfrac»«mi»distance«/mi»«mi»time«/mi»«/mfrac»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»and«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»velocity«/mi»«mo»=«/mo»«mfrac»«mi»displacement«/mi»«mi»time«/mi»«/mfrac»«/math»
Note distance and displacement are equivalent when an object travels in a straight line without any changes in direction. Therefore, speed and velocity are equivalent under the same conditions.
An object is moved «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»6«/mn»«mo»§#160;«/mo»«mi»cm«/mi»«/math» forward along a straight line a distance, and then it is moved «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mi»cm«/mi»«/math» backward.
a.
Find the total distance travelled by the object and find the displacement of the object.
b.
Find the speed and the velocity of the object if it was moving for «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math».
a.
The total distance travelled is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»6«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»cm«/mi»«mo»+«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»cm«/mi»«mo»=«/mo»«mn»8«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»cm«/mi»«/math».
The displacement is calculated by finding how far the object is from the starting point. Since the object moved «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi»cm«/mi»«/math» backward from its initial forward movement of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»6«/mn»«mo»§#160;«/mo»«mi»cm«/mi»«/math», the object is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»6«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»cm«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»cm«/mi»«mo»=«/mo»«mn»4«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»cm«/mi»«/math» from its the starting point.
The displacement is calculated by finding how far the object is from the starting point. Since the object moved «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»2«/mn»«mo»§#160;«/mo»«mi»cm«/mi»«/math» backward from its initial forward movement of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»6«/mn»«mo»§#160;«/mo»«mi»cm«/mi»«/math», the object is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»6«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»cm«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#8722;«/mo»«mn»2«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»cm«/mi»«mo»=«/mo»«mn»4«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»cm«/mi»«/math» from its the starting point.
b.
Calculate speed and velocity using the formulas presented above.
Note: The term average is used because speed and velocity are seldom truly uniform for more than a very short period of time.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left center center center center center right center left¨»«mtr»«mtd»«mi»average«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»speed«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»distance«/mi»«mi»time«/mi»«/mfrac»«/mtd»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«mtd»«mi»average«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»velocity«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»displacement«/mi»«mi»time«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»8«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»cm«/mi»«/mrow»«mrow»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»s«/mi»«/mrow»«/mfrac»«/mtd»«mtd/»«mtd/»«mtd»«mi»and«/mi»«/mtd»«mtd/»«mtd/»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mn»4«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»cm«/mi»«/mrow»«mrow»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»s«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»cm«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/mtd»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»cm«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/mtd»«/mtr»«/mtable»«/math»
Note: The term average is used because speed and velocity are seldom truly uniform for more than a very short period of time.