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# Ap Physics C Mathematical Concepts Vectors

## on Aug 19, 2009

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## Ap Physics C Mathematical Concepts VectorsPresentation Transcript

• Mathematical Concepts: Polynomials, Trigonometry and Vectors AP Physics C 20 Aug 2009
• Polynomials review
• “ zero order” f(x) = m x 0
• “ linear”: f(x) = mx 1 +b
• “ quadratic”: f(x) = mx 2 + nx 1 + b
• And so on….
• Inverse functions
• Inverse
• Inverse square
• Polynomial graphs Linear Quadratic Inverse Inverse Square
• Right triangle trig
• Trigonometry is merely definitions and relationships.
• Starts with the right triangle.
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 Trig functions Reciprocal trig functions Reciprocal trig functions Trig identities
• 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 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 addition graphical 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
• 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 N
• 57.3 m, 36 S of W
• 17.4 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 Multiplication Dot Product
• The dot product (or scalar product), is denoted by:
• It is the projection of vector A multiplied by the magnitude of vector B.
• Vector multiplication Dot product
• In terms of components, the dot product can be determined by the following:
• Vector multiplication Dot product Example #1
• Find the scalar product of the following two vectors. A has a magnitude of 4, B has a magnitude of 5.
53 º 50 º A B
• Vector Multiplication Dot Product Example #2
• Find the angle between the two vectors
• Vector Multiplication Cross 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.
• Vector multiplication Cross product (direction)
• Vector Multiplication Cross product
• The vector C represents the solution to the cross product of A and B .
• To find the components of C, use the following
• Vector Multiplication Cross product
• This is more easily remembered using a determinant
• Vector Multiplication Cross 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 30 º with the + x-axis. What is the cross product of these two vectors?