L1 Distance, Displacement, and Velocity - Part 8
Completion requirements
Unit 7A
Integrals Part 1
Lesson 7: Position and Velocity
Summary
When working with the concepts in this Lesson, it is helpful to follow a set of general steps.After reading the problem carefully, decide if it is asking for an average rate of change or an instantaneous rate of change. For instance, in problems involving average velocity, a time interval will be given. In problems involving instantaneous
velocity, a single instant in time will be specified.
Make a note of any algebraic signs for the quantities found. Signs indicate direction of motion: up and right are positive, and left and down are negative.
Express all answers with appropriate units. The position of an object is given as a unit of length and velocity is measured as a rate, such as «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cm«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math».
Average rate of change is different from instantaneous rate of change.
For «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math»:
| Average Rate of Change
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Instantaneous Rate of Change |
| «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mo»§#916;«/mo»«mi»y«/mi»«/mrow»«mrow»«mo»§#916;«/mo»«mi»x«/mi»«/mrow»«/mfrac»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mfrac»«mrow»«mo»§#916;«/mo»«mi»f«/mi»«/mrow»«mrow»«mo»§#916;«/mo»«mi»x«/mi»«/mrow»«/mfrac»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»f«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/math» |
| slope of a secant of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math»
|
slope of a tangent to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math»
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The following are the specific formulas used for average and instantaneous rates of change of velocity.
| Average Velocity
|
Instantaneous Velocity |
| «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»v«/mi»«mrow»«mi»a«/mi»«mi»v«/mi»«mi»e«/mi»«/mrow»«/msub»«mo»=«/mo»«mfrac»«mrow»«mo»§#916;«/mo»«mi»s«/mi»«/mrow»«mrow»«mo»§#916;«/mo»«mi»t«/mi»«/mrow»«/mfrac»«/math»
|
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/math»
|
| slope of a secant of the position-time graph
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slope of a tangent to the position-time graph
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Since the velocity function is the derivative of the position function (displacement), the object reaches its local maximum or minimum displacement when the velocity is zero. This is represented graphically by a horizontal tangent line.