mathematical concepts: polynomials, trigonometry and vectors ap physics c 20 aug 2009
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Mathematical Concepts:Polynomials, Trigonometry and Vectors
AP Physics C
20 Aug 2009
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
Polynomial graphs
Linear
Quadratic
Inverse
InverseSquare
Right triangle trig
Trigonometry is merely definitions and relationships. Starts with the right triangle.
b
ac
bc
a
tan
cos
sin
a
b
c
Special Right Triangles
30-60-90 triangles 45-45-90 triangles 37-53-90 triangles (3-4-5 triangles)
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
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, ….
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
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.
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.
Vector additiongraphical method
+ =
+ =
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
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?
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
Vector multiplicationDot product
In terms of components, the dot product can be determined by the following:
zzyyxx BABABABA
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
Vector MultiplicationDot Product Example #2
Find the angle between the two vectors
kjiB
kjiA
ˆˆ2ˆ4
ˆˆ3ˆ2
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
Vector multiplicationCross product (direction)
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
Vector MultiplicationCross product
This is more easily remembered using a determinant
zyx
zyx
BBB
AAA
kji
BA
ˆˆˆ
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?
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