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Mathematical Concepts:Polynomials, Trigonometry and Vectors
AP Physics C
20 Aug 2009
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Polynomials review
“zero order” f(x) = mx0
“linear”: f(x) = mx1 +b “quadratic”: f(x) = mx2 + nx1 + b And so on…. Inverse functions
Inverse
Inverse square
x
axf
2x
axf
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Polynomial graphs
Linear
Quadratic
Inverse
InverseSquare
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Right triangle trig
Trigonometry is merely definitions and relationships. Starts with the right triangle.
b
ac
bc
a
tan
cos
sin
a
b
c
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Special Right Triangles
30-60-90 triangles 45-45-90 triangles 37-53-90 triangles (3-4-5 triangles)
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Trigonometric functions & identities
x
x
x
tan
cos
sin
xx
xx
xx
tan
1cot
cos
1sec
sin
1csc
yxxy
yxxy
yxxy
1
1
1
cottan
coscos
sinsin
Trig functionsReciprocal trig
functionsReciprocal trig
functions
Trig identities
x
xxcos
sintan xx 22 cossin1
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Vectors
A vector is a quantity that has both a direction and a scalar Force, velocity, acceleration, momentum,
impulse, displacement, torque, …. A scalar is a quanitiy that has only a
magnitude Mass, distance, speed, energy, ….
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Cartesian coordinate system
r x x y y z za a a
r x x y y z za a a
r x i y j z ka a a
or
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Resolving a 2-d vector
“Unresolved” vectors are given by a magnitude and an angle from some reference point. Break the vector up into components by
creating a right triangle. The magnitude is the length of the
hypotenuse of the triangle.
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Resolving a 2-d vector (example #1)
A projectile is launched from the ground at an angle of 30 degrees traveling at a speed of 500 m/s. Resolve the velocity vector into x and y components.
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Vector additiongraphical method
+ =
+ =
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Vector additionnumerical method
Add each component of the vector separately. The sum is the value of the vector in a
particular direction. Subtracting vectors? To get the vector into “magnitude and
angle” format, reverse the process
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Vector addition example #1
Three contestants of a game show are brought to the center of a large, flat field. Each is given a compass, a shovel, a meter stick, and the following directions:
72.4 m, 32 E of N57.3 m, 36 S of W17.4 m, S
The three displacements are the directions to where the keys to a new Porche are buried. Two contestants start measuring, but the winner first calculates where to go. Why? What is the result of her calculation?
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Vector MultiplicationDot Product
The dot product (or scalar product), is denoted by:
It is the projection of vector A multiplied by the magnitude of vector B.
cosBABA
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Vector multiplicationDot product
In terms of components, the dot product can be determined by the following:
zzyyxx BABABABA
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Vector multiplicationDot product Example #1
Find the scalar product of the following two vectors. A has a magnitude of 4, B has a magnitude of 5.
53º50º
A
B
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Vector MultiplicationDot Product Example #2
Find the angle between the two vectors
kjiB
kjiA
ˆˆ2ˆ4
ˆˆ3ˆ2
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Vector MultiplicationCross Product (magnitude)
The cross product is a way to multiply 2 vectors and get a third vector as an answer.
The cross product is denoted by:
The magnitude of the cross product is the product of the magnitude of B and the component of A perpendicular to B.
sinBACBA
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Vector multiplicationCross product (direction)
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Vector MultiplicationCross product
The vector C represents the solution to the cross product of A and B.
To find the components of C, use the following
xyyxZ
zxxzy
yzZyx
BABAC
BABAC
BABAC
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Vector MultiplicationCross product
This is more easily remembered using a determinant
zyx
zyx
BBB
AAA
kji
BA
ˆˆˆ
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Vector MultiplicationCross Product Example #1
Vector A has a magnitude of 6 units and is in the direction of the + x-axis. Vector B has a magnitude of 4 units and lies in the x-y plane, making an angle of 30º with the + x-axis. What is the cross product of these two vectors?