L1 Distance, Displacement, and Velocity - Part 4
Completion requirements
Unit 7B
Integrals Part 2
Lesson 1: Distance, Displacement, and Velocity
A function «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» that defines the position of an object, relative to the origin, as a function of time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math»
is called a position function. If the object changes its position by the amount «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#916;«/mo»«mi»s«/mi»«/math» over a period of time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#916;«/mo»«mi»t«/mi»«/math»,
the ratio for velocity is as follows.
The velocity of a moving object at any instant is the first derivative, with the respect to time, of the displacement function. A positive velocity indicates motion to the right or upward, and a negative velocity indicates motion to the left or downward relative to a fixed point.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»velocity«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»change«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi»in«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»position«/mi»«/mrow»«mrow»«mi»change«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»in«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»time«/mi»«/mrow»«/mfrac»«mo mathvariant=¨italic¨»=«/mo»«mfrac»«mrow»«mo mathvariant=¨italic¨»§#916;«/mo»«mi»s«/mi»«/mrow»«mrow»«mo mathvariant=¨italic¨»§#916;«/mo»«mi»t«/mi»«/mrow»«/mfrac»«/math»
The velocity of a moving object at any instant is the first derivative, with the respect to time, of the displacement function. A positive velocity indicates motion to the right or upward, and a negative velocity indicates motion to the left or downward relative to a fixed point.
If a rock is thrown vertically upward from ground level with an initial velocity of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»30«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math»,
it will stop travelling upward at some point and fall back toward the ground. The height «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math», in metres, of the rock is a function of time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«/math»,
in seconds, and is given by «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»30«/mn»«mi»t«/mi»«/math», where «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»§#8805;«/mo»«mn»0«/mn»«/math».
Determine
The displacement is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»15«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math».
The average velocity is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math».
Determine
a.
the initial position of the rock,
b.
the displacement from time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»1«/mn»«/math» to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»4«/mn»«/math»,
c.
the average velocity from «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»1«/mn»«/math» to «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»4«/mn»«/math»,
d.
the instantaneous velocity at time «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»4«/mn»«/math», and
e.
The rock’s maximum height and the time it takes to reach the maximum height.
a.
The initial position of the rock occurs when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»0«/mn»«/math».
The initial position of the rock is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»0«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»s«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»30«/mn»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»s«/mi»«mfenced»«mn»0«/mn»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mfenced»«mn»0«/mn»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»30«/mn»«mfenced»«mn»0«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/math»
The initial position of the rock is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»0«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math».
b.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»displacement«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»s«/mi»«mfenced»«mn»4«/mn»«/mfenced»«mo»§#8722;«/mo»«mi»s«/mi»«mfenced»«mn»1«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfenced open=¨[¨ close=¨]¨»«mrow»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mfenced»«mn»4«/mn»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»30«/mn»«mfenced»«mn»4«/mn»«/mfenced»«/mrow»«/mfenced»«mo»§#8722;«/mo»«mfenced open=¨[¨ close=¨]¨»«mrow»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mfenced»«mn»1«/mn»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»30«/mn»«mfenced»«mn»1«/mn»«/mfenced»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»40«/mn»«mo»§#8722;«/mo»«mn»25«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»15«/mn»«/mtd»«/mtr»«/mtable»«/math»
The displacement is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»15«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math».
c.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msub»«mi»v«/mi»«mrow»«mi»a«/mi»«mi»v«/mi»«mi»e«/mi»«/mrow»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»displacement«/mi»«mrow»«mi»change«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»in«/mi»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»time«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»15«/mn»«mn»3«/mn»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«/mtd»«/mtr»«/mtable»«/math»
The average velocity is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»5«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math».
d.
Instantaneous velocity is calculated using the derivative of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»s«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math».
Find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»4«/mn»«/math».
The rock is falling at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»s«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»s«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»30«/mn»«mi»t«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«mi»d«/mi»«mi»s«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»t«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»10«/mn»«mi»t«/mi»«mo»+«/mo»«mn»30«/mn»«/mtd»«/mtr»«/mtable»«/math»
Find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»4«/mn»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»10«/mn»«mfenced»«mn»4«/mn»«/mfenced»«mo»+«/mo»«mn»30«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»10«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/mtd»«/mtr»«/mtable»«/math»
The rock is falling at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«mo»/«/mo»«mi mathvariant=¨normal¨»s«/mi»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»4«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»s«/mi»«/math».
e.
The rock reaches its maximum height when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«mo»=«/mo»«mn»0«/mn»«/math».
Find «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»s«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»3«/mn»«/math».
The rock reaches a maximum height of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»45«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»s«/mi»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»v«/mi»«mfenced»«mi»t«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»10«/mn»«mi»t«/mi»«mo»+«/mo»«mn»30«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»0«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»10«/mn»«mi»t«/mi»«mo»+«/mo»«mn»30«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»10«/mn»«mi»t«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»30«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»t«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«/mtd»«/mtr»«/mtable»«/math»
| The rock is at a maximum height at the instant the velocity is zero because it is at this instant when the rock changes direction and begins to fall. |
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»s«/mi»«mfenced»«mn»3«/mn»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»5«/mn»«msup»«mfenced»«mn»3«/mn»«/mfenced»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»30«/mn»«mfenced»«mn»3«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#8722;«/mo»«mn»45«/mn»«mo»+«/mo»«mn»90«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»45«/mn»«/mtd»«/mtr»«/mtable»«/math»
The rock reaches a maximum height of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mn»45«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/math» when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»t«/mi»«mo»=«/mo»«mn»3«/mn»«mi mathvariant=¨normal¨» «/mi»«mo»§#160;«/mo»«mi»s«/mi»«/math».