6.3 parametric equations and motion › nhhs › math › fcierech › files › 2012 › 08 ›...
TRANSCRIPT
![Page 1: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/1.jpg)
6.3 Parametric Equations and Motion
![Page 2: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/2.jpg)
I. Definition
A.) Parametric Curve – Graph of ordered pairs (x, y) where x = f(t) and y = f(t).
B.) Parametric Equations– Any equation in the form of x = f(t) and y = f(t). t is called the parameter on an interval I.
![Page 3: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/3.jpg)
C.) Ex. 1– “Old School” – A ball is thrown in the air with an initial velocity of 32 ft./sec from a height of 6 ft. Write an equation to model the height of the ball as a function of time.
2( ) 16 32 6h t t t
Let’s look at the graph
![Page 4: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/4.jpg)
(0,6)
(1,22)
(2,6)
![Page 5: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/5.jpg)
If the ball is thrown directly upward, we can model the position of the ball using two different functions of t.
The horizontal position of the ball is now on a vertical axis, as it is in “real-life”. While the graph models its vertical position with respect to time, we can also see the velocity and the acceleration by changing our mode to parametric and our line to -0
216 32 6y t t 2x
![Page 6: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/6.jpg)
A.) Ex. 2- Graph the following parametric equation by hand.
Make a table:
II. Graphing by Hand
2 2 3 for 2 3
x ty t t
t x y-2-10123
![Page 7: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/7.jpg)
t x y
-2 2 -6
-1 -1 -3
0 -2 0
1 -1 3
2 2 6
3 7 9
![Page 8: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/8.jpg)
III. Eliminating the Parameter
A.) Ex. 3- Determine the graph to the parametric equation by eliminating the parameter.
2 33
2
x txt
2
2 3 for 3 3
x ty t t
3 9x
2
2
32
1 3 9- 4 2 4
xy
y x x
![Page 9: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/9.jpg)
B.) Ex. 4- Determine the graph to the parametric equation by eliminating the parameter.
1cost x
cos sinx t y t
1sin cos y x
Very unfamiliar and hard to work with!!!
![Page 10: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/10.jpg)
2 2 2 2cos sinx y t t
2 2
2 2
cos sin
x ty t
2 2 1x y
The circle centered at the origin with a radius of 1
What if we squared both sides of both equations?
Now, add the equations!
![Page 11: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/11.jpg)
A.) Ex. 5- Find the parameterization for the line through the points (-3, -3) and (5, 1).
IV. Lines and Line Segments
A
B
![Page 12: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/12.jpg)
Let P(x, y) be a point on the line through A and B and let O be the origin.
O
A
B
P
The vector
is a scalar multiple of the
vector OB OA AB
OP OA AP
![Page 13: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/13.jpg)
OP OA t OB OA
, 3, 3 5,1 OP x y OA OB
3, 3 8, 4x y t
3 8 3 4x t y t
t
8 3 4 3x t y t
![Page 14: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/14.jpg)
B.) Ex. 6 - Find the parameterization for the line through the points A=(3, 4) and B=(6, -3).
OP OA t OB OA
, 3, 4 6, 3 OP x y OA OB
3, 4 3, 7x y t
3 3 4 7x t y t
t 3 3 7 4x t y t
![Page 15: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/15.jpg)
What if we wanted the line segment with endpoints Aand B?
Which values of t produce these endpoints?
0 1t t
0 1t
3 3 3 3 3 6t t
![Page 16: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/16.jpg)
V. Simulating Motion
A.) A particle moves along a horizontal line so that the position s (in feet) at any time t ≥ 0 seconds is given by the function
B.) Ex. 7- Estimate the values at which the particle changes direction given
3( ) 8 1.s t t t
3 3.t
![Page 17: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/17.jpg)
Graph it. Place the equation in X1T and choose an arbitrary Y1T value.
What kind of window do we need?
Trace it.
31
1
8 14
T
T
X T TY
:[ 3,3] :[ 10,50] :[0,6]T X Y
1.637t
![Page 18: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/18.jpg)
Another way: Let Y1T = T.
Window:
Now we can visually see when the particle changes direction!
31
1
8 1T
T
X T TY T
:[ 3,3] :[ 10,50] :[ 10,10]T X Y
![Page 19: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/19.jpg)
VI. Projectile Motion
A.) Not everything is straight vertically or horizontally. Take for example, hitting a baseball. There is both a horizontal and vertical component. The velocity vector is
The path that the object takes can be modeled by the parametric equations
0 0cos , sinv v v
2
0
cos
16 sino
o
x v t
y t v t y
Horizontal Distance
Vertical Position
![Page 20: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/20.jpg)
B.) Ex. 8- Sally hits a softball 2.5 feet above the plate with an initial speed of 110 feet per second at an angle of 22ºwith the horizontal. Will the ball clear the 5 ft. fence 250 feet away?
First, determine the equations for the ball:
Now, determine the window
Now, graph and trace to see if it clears the wall
1
21
110cos 22
16 110sin 22 2.5T
T
X t
Y t t
:[0,5] step : .1 :[0, 280] :[0,30]T T X Y
![Page 21: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/21.jpg)
C.) Determine the equations for the wall:
Change your MODE to Simultaneous
Now, just graph to see if it clears the wall
2
2
250T
T
XY T
D.) General the equation for the wall:
2
2
DistanceWall's height
max
T
T
XT
YT
![Page 22: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/22.jpg)
VII. Modeling Ferris Wheels
A.) Ex. 9 - A Ferris wheel with a radius of 20 feet makes 1 complete revolution every 10 seconds. The lowest point of the Ferris wheel is 5 feet above the ground. Determine the parametric equations which will model the height of a rider starting in the 3 o’clock position at t = 0.
![Page 23: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/23.jpg)
1.) Center the Ferris wheel on the vertical axis such that the center will be at the point (0, 25).
We know, from Chapter 5 that
But, θ must be in terms of t. Since it takes 10 sec. to complete 1 revolution,
20cos25 20sin
xy
2 radians radians10 sec 5 sec
![Page 24: 6.3 Parametric Equations and Motion › nhhs › math › fcierech › files › 2012 › 08 › ... · VI. Projectile Motion A.) Not everything is straight vertically or horizontally](https://reader033.vdocument.in/reader033/viewer/2022052722/5f0c13de7e708231d433a1eb/html5/thumbnails/24.jpg)
Therefore,
and
1
1
20cos 36
25 20sin 36T
T
X t
Y t
1
1
20cos5
25 20sin5
T
T
X t
Y t
5t