Ap Physics C Mathematical Concepts Vectors


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Ap Physics C Mathematical Concepts Vectors

  1. 1. Mathematical Concepts: Polynomials, Trigonometry and Vectors AP Physics C 20 Aug 2009
  2. 2. Polynomials review <ul><li>“ zero order” f(x) = m x 0 </li></ul><ul><li>“ linear”: f(x) = mx 1 +b </li></ul><ul><li>“ quadratic”: f(x) = mx 2 + nx 1 + b </li></ul><ul><li>And so on…. </li></ul><ul><li>Inverse functions </li></ul><ul><ul><li>Inverse </li></ul></ul><ul><ul><li>Inverse square </li></ul></ul>
  3. 3. Polynomial graphs Linear Quadratic Inverse Inverse Square
  4. 4. Right triangle trig <ul><li>Trigonometry is merely definitions and relationships. </li></ul><ul><ul><li>Starts with the right triangle. </li></ul></ul>a b c 
  5. 5. Special Right Triangles <ul><li>30-60-90 triangles </li></ul><ul><li>45-45-90 triangles </li></ul><ul><li>37-53-90 triangles (3-4-5 triangles) </li></ul>
  6. 6. Trigonometric functions & identities Trig functions Reciprocal trig functions Reciprocal trig functions Trig identities
  7. 7. Vectors <ul><li>A vector is a quantity that has both a direction and a scalar </li></ul><ul><ul><li>Force, velocity, acceleration, momentum, impulse, displacement, torque, …. </li></ul></ul><ul><li>A scalar is a quanitiy that has only a magnitude </li></ul><ul><ul><li>Mass, distance, speed, energy, …. </li></ul></ul>
  8. 8. Cartesian coordinate system or
  9. 9. Resolving a 2-d vector <ul><li>“Unresolved” vectors are given by a magnitude and an angle from some reference point. </li></ul><ul><ul><li>Break the vector up into components by creating a right triangle. </li></ul></ul><ul><ul><li>The magnitude is the length of the hypotenuse of the triangle. </li></ul></ul>
  10. 10. Resolving a 2-d vector (example #1) <ul><li>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. </li></ul>
  11. 11. Vector addition graphical method + = + =
  12. 12. Vector addition numerical method <ul><li>Add each component of the vector separately. </li></ul><ul><ul><li>The sum is the value of the vector in a particular direction. </li></ul></ul><ul><li>Subtracting vectors? </li></ul><ul><li>To get the vector into “magnitude and angle” format, reverse the process </li></ul>
  13. 13. Vector addition example #1 <ul><li>Three contestants of a game show are brought to the </li></ul><ul><li>center of a large, flat field. Each is given a compass, a </li></ul><ul><li>shovel, a meter stick, and the following directions: </li></ul><ul><li>72.4 m, 32 E of N </li></ul><ul><li>57.3 m, 36 S of W </li></ul><ul><li>17.4 m, S </li></ul><ul><li>The three displacements are the directions to where </li></ul><ul><li>the keys to a new Porche are buried. Two contestants </li></ul><ul><li>start measuring, but the winner first calculates where to </li></ul><ul><li>go. Why? What is the result of her calculation? </li></ul>
  14. 14. Vector Multiplication Dot Product <ul><li>The dot product (or scalar product), is denoted by: </li></ul><ul><li>It is the projection of vector A multiplied by the magnitude of vector B. </li></ul>
  15. 15. Vector multiplication Dot product <ul><li>In terms of components, the dot product can be determined by the following: </li></ul>
  16. 16. Vector multiplication Dot product Example #1 <ul><li>Find the scalar product of the following two vectors. A has a magnitude of 4, B has a magnitude of 5. </li></ul>53 º 50 º A B
  17. 17. Vector Multiplication Dot Product Example #2 <ul><li>Find the angle between the two vectors </li></ul>
  18. 18. Vector Multiplication Cross Product (magnitude) <ul><li>The cross product is a way to multiply 2 vectors and get a third vector as an answer. </li></ul><ul><li>The cross product is denoted by: </li></ul><ul><li>The magnitude of the cross product is the product of the magnitude of B and the component of A perpendicular to B. </li></ul>
  19. 19. Vector multiplication Cross product (direction)
  20. 20. Vector Multiplication Cross product <ul><li>The vector C represents the solution to the cross product of A and B . </li></ul><ul><li>To find the components of C, use the following </li></ul>
  21. 21. Vector Multiplication Cross product <ul><li>This is more easily remembered using a determinant </li></ul>
  22. 22. Vector Multiplication Cross Product Example #1 <ul><li>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? </li></ul>