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Ap Physics C Mathematical Concepts Vectors
 

Ap Physics C Mathematical Concepts Vectors

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    Ap Physics C Mathematical Concepts Vectors Ap Physics C Mathematical Concepts Vectors Presentation 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 + = + =
    • Vector addition numerical 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 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?