Transcript A Mathematics Review

A Mathematics Review
Unit 1 Presentation 2
Why Review?
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Mathematics are a very important
part of Physics
Graphing, Trigonometry, and
Algebraic concepts are used often
Solving equations and breaking
down vectors are two important
skills
Graphing Review
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Graphing in Physics done on a
Cartesian Coordinate System
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Also known as an x-y plane
Can also graph in Polar Coordinates
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Also known as an r,q plane
Very Useful in Vector Analysis
Rectangular vs. Polar Coordinates
Rectangular Coordinate
System
X-Y Axes Present (dark
black lines)
X Variable
Y Variable
Polar Coordinate System
NO X-Y Axes
R Variable (red lines)
q Variable (blue/black
lines)
Trigonometry Review
Remember
SOHCAHTOA?
Opposite Side
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opposite
hypotenuse
adjacent
cos q 
hypotenuse
opposite sin q
tan q 

adjacent cos q
sin q 
Pythagorean Theorem for Right Triangles:
a b  c
2
q
Adjacent Side
2
2
Using Polar Coordinates
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To convert from Rectangular to Polar
Coordinates (or vice versa), use the following:
r x y
2
2
 y
q  tan  
x
1
x  r cos q
y  r sin q
Polar Coordinates Example
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Convert (-3.50 m, -2.50 m) from
Cartesian coordinates to Polar
coordinates.
x  3.50m r  x 2  y 2
r  (3.50) 2  (2.50) 2  4.30m
y  2.50m
1  y 
q  tan  
1   2.50 
  36
 x  q  tan 
  3.50 
But, consider a displacement in the negative x and y directions.
That is in Quadrant III, so, since polar coordinates start with the
Positive x axis, we must add 180° to our answer, giving us a final
answer of 216°
Another Polar Coordinates Example
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Convert 12m @ 75 degrees into x and y coordinates.
First, consider that this displacement is in Quadrant I, so
our answers for x and y should both be positive.
r  12m
q  75
x  r cosq
x  12  cos(75)  3.11m
y  r sin q
y  12  sin( 75)  11.60m
Trigonometry Review
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Calculate the height of a building if you can see the top of
the building at an angle of 39.0° and 46.0 m away from
its base.
39.0°
Building Height
First, draw a picture.
Since we know the adjacent side and
want to find the opposite side, we should
use the tangent ratio.
46.0 m
q  39.0
adjacent  46.0m
opposite  ?
tan q 
opposite
adjacent
tan q  adjacent  opposite
tan( 39.0)  46.0m  37.3m
Another Trigonometry Example
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An airplane travels 4.50 x 102 km due east and then
travels an unknown distance due north. Finally, it returns
to its starting point by traveling a distance of 525 km.
How far did the airplane travel in the northerly direction?
First, draw a picture.
x km
N
450 km
a  450km
b?
c  525km
This problem would best be solved
using the Pythagorean Theorem.
a2  b2  c2
b  c2  a2
b  (525) 2  (450) 2  270km