L2 Displacement, Velocity, and Acceleration - Part 9
Completion requirements
Unit 7B
Integrals Part 2
Lesson 2: Displacement, Velocity and Acceleration
Summary
When working with the concepts in this Lesson, it is helpful to follow a set of general steps.Click each coloured tab to view each step.
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 that require an average rate of change, a time interval will be given. For problems that require
an instantaneous rate of change, a single moment in time will be specified.
Make a note of the signs of the quantities found. Signs indicate direction of motion: up and right are positive, left and down are negative. For an object to speed up, its instantaneous velocity and acceleration must be in the same direction (both
positive or both negative); for an object to slow down, instantaneous velocity and acceleration are in opposite directions (one positive, the other negative).
| Velocity | Acceleration | Speed |
| «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»+«/mo»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»+«/mo»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»+«/mo»«/math» (increasing)
|
| «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»+«/mo»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«/math»(decreasing)
|
| «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»+«/mo»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«/math» (decreasing)
|
| «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»+«/mo»«/math» (increasing)
|
Express all answers with appropriate units. The position of an object is given as a unit of length, velocity is given as a length per unit of time (eg., «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cm«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math»), and acceleration is given as a velocity per unit of time (eg., «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»cm«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«mo»=«/mo»«mi»cm«/mi»«mo»/«/mo»«msup»«mi mathvariant=¨normal¨»s«/mi»«mn»2«/mn»«/msup»«/math»).
The following are the specific formulas used for average and instantaneous rates of change for 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 |
slope of a tangent to the position-time graph
|
The following are the specific formulas used for average and instantaneous rates of change for acceleration.
| Average Velocity
|
Instantaneous Velocity |
| «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi»a«/mi»«mrow»«mi»a«/mi»«mi»v«/mi»«mi»e«/mi»«/mrow»«/msub»«mo»=«/mo»«mfrac»«mrow»«mo»§#916;«/mo»«mi»v«/mi»«/mrow»«mrow»«mo»§#916;«/mo»«mi»t«/mi»«/mrow»«/mfrac»«/math» |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»a«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»v«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«msup»«mi»d«/mi»«mn»2«/mn»«/msup»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/math» |
| slope of a secant of the velocity-time graph |
slope of a tangent to the velocity-time graph
|
When velocity is zero, the object reaches its local maximum or minimum position, which is represented by a horizontal tangent on the graph of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»s«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math». When acceleration is zero, the object reaches its maximum or minimum velocity, which is represented by a horizontal tangent on the graph of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math». When «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»a«/mi»«mo»=«/mo»«mn»0«/mn»«/math», velocity is constant.