Mathematical Concepts: Polynomials, Trigonometry and Vectors AP Physics C 20 Aug 2009

Polynomials review “zero order” f(x) = mx 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 Square

Right triangle trig Trigonometry is merely definitions and relationships.  Starts with the right triangle. a b c 

Special Right Triangles triangles triangles 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. 53º 50º 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 30º with the + x-axis. What is the cross product of these two vectors?