Parametric Equations for Moving Objects

Property: Parametric Equations of a Cycloid

x = a(t-sint)
y = a(1-cost)

t is the number of radians the wheel has rolled so far
a is the radius

Objective : Be able to find the parametric equations for the path of a moving object.
Here is a example to explain how to do this:

An airplane is flying to LA from Nevada at a speed of 400 mi/hr. It is at point (20,30) on a Cartesian plane at t = 0 hr. LA is the origin. There are two winds going in different directions blowing. The wind blowing south is moving at a velocity of 90 mi/hr. The wind blowing west is moving at a velocity of 120 mi/hr.

Find the parametric equations for the airplane's path. Use t as hours.

x=20-120t
Since the original point of the plane is (20,30) and the velocity of the force blowing horizontally is 120mi/hr, x = the orignial x value + distance (which is rate x time)
In the same way
y=30-90t
The distance is negative becuase the wind is blowing in the negative directions of x and y.

Predict how long it will take for the plane to be 5 miles north of LA.
First plug 5 in for y.
5=30+90t
t=1.66666667
It would take 1.667 hours to get to this point.

try this site for more help: http://www.assembleme.com/post_2004_07_22_parametric_equations.pdf

i love the yeah yeah yeahs...http://youtube.com/watch?v=1UiNr8T2Mrc. -turn into

Anna you're next.

Thursday's Test Topics

Here’s a list of topics for Thursday’s test:

Precalculus Chapter 13 Test Topics:
Relationships between polar and Cartesian coordinates
Plot polar points.
Convert polar equation to Cartesian
Transform a polar graph/equation
Parametric equations – linear motion (word problem)
Determine a point on a curve given polar data
Describe the formation/creation of a polar curve
Parametric equations – nonlinear motion (word problems)

I’ve posted the solutions to the review handout on the class website .
See you in class! I’ll be around until 3:00 pm on Wednesday afternoon, and in early on Thursday.

If you have built castles in the air, that is where they should be; now put foundations under them.
- Henry David Thoreau

13-3: Intersections of Polar Curves

Hi Guys! Sorry it took so long for this post to get online, but I wasn't 100% sure if it was my turn or not. I'm just going to assume it is. As usual, a bland, colorless blog...

The main focus of this chapter is on determining whether or not an intersection of two graphs in polar coordinates is a true intersection.

Here we see a graph with the polar equations

3 + 3 cos (theta)
and
5 sin (2 theta)



There appear to be eight intersections, as shown by the black dots (two are kind of smushed together), but not all are TRUE intersections. In order for an intersection to be true, the two points must meet at the same theta value. In order to find out which values are true, go to the "Mode" function on your calculator, and change the standard "Sequential" setting to "Simul" (which is Latin for "at the same time"- just a little fun fact.) If you visit your graph once again, the two equations will be graphed simultaneously (hence the "simul" setting). Watch closely for the intersections- the true intersections of the graphs will meet at the same point at the same time. After graphing our equations using "simul", we find that only these points are true intersections.



In order to find the coordinates of these points, we must leave polar mode and switch back to "function" mode. Plug in the exact same points in function mode, but using "x" as opposed to "theta". Find the intersections between the two graphs using the "calc" function; these points represent to coordinates on the polar graph.

Here is the graph in function (and radian) mode...



Here's a Problem!

Graph the following equations, and indicate the true intersections. calculate the values.

r= 2 + 8cos(theta)
r= 12 sin (3theta)

Solution

Start by graphing the two equations in polar, simultaneous mode. Make sure your window is big enough to see the intersections. Your graphs should look something like this...



By using the "SIMUL" function, you should end up with about these points as "true" intersections.



Plugging the values into the "function" mode gives you a graph like this-



The values for these points are (17.760, 9.619), (47.267, 7.431), (117.329, -1.673)



Additional Help




Debby's Up Next!




Personalization:
There's a Sketch Comedy group called The Whitest Kids You Know who recently got a show on fuse- these guys are extremely talented and have a lot of creative, original ideas. Here are some of their videos.














The Girl in this video is my cousin Sarah- this was the first sketch of the first episode of the show.



This Video is by a comedy group called "Derrick"

Thursday's Quiz

Here’s a list of topics for Thursday’s quiz:

Precalculus Quiz 13.1-3 Topics
Plotting polar coordinates
Multiple representations of polar coordinates
Plotting a curve given a table of polar coordinates
Interpreting/Reading a polar graph
Determining the equation of a polar graph
True/False intersections of polar graphs
Converting a polar equation to Cartesian

That’s it! The format’s the same as always – ½ Non-calculator, ½ Calculator. There are 7 questions on the Non-calculator portion and 9 questions on the Calculator section. I’ll be around after school on Wednesday, and on campus by 8:00 AM on Thursday.

People are like stained-glass windows. They sparkle and shine when the sun is out, but when the darkness sets in, their true beauty is revealed only if there is a light from within.
—PHYSICIAN ELISABETH KÜBLER-ROSS

My job:

answers to chapter 10 review questions!

hey everybody!

i didn't put the answers on the review sheet, but here they are.

1. square root of 305.

2. 5 root 10 or square root of 250

3. 4i - 11j -13k

4. square root of 306

5.

6. 23

7. 43i + 29j -49k

8. cos alpha = 7/root 102, cos beta = -7/root 102, cos gamma = 2/root 102

9. 3i + 12j + 3k

10. 8i + 43j + 6k

11. dot product of a and b = 15, so the angle = 84.61

12. -2i - 2j - 2k

i hope these are helpful (and right)

-claire, nat, and allison

Thursday's Test Topics

Here’s a list of topics for Thursday’s test:

Precalculus Chapter 10 Test Topics:
Unit vectors
Planes – Standard Equation vs. normal vectors and points
Points on (or not on) a plane
Lines in Space – points and unit vectors
Direction Angles
Direction Cosines – properties
Vectors between two points
Angles between vectors
Scalar and vector projections
Cross products

I’ve posted a sample test and review on the class website which I’ll distribute in class tomorrow. Review 10-d will not be as helpful as 10-b, 10-c and 10-e.

See you in class!

Here’s a little something that shows the beauty of mathematics:

Math Art: The Beauty and Symmetry of Mathematics.

1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321

1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 +10= 1111111111

9 x 9 + 7 = 88
98! x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 x 9 + 0 = 888888888

Brilliant, isn't it? And finally, take a look at this symmetry:

1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321

A unique approach to homework: