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Vectors

Vectors. Overview (Holt p82). Motion Progression (1D  2D). Started with 1 dimensional (1D) motion Motion along X or Y axis Now consider 2 dimensional motion (2D), that is between the Y and X axes

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Vectors

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  1. Vectors Overview (Holt p82)

  2. Motion Progression (1D  2D) • Started with 1 dimensional (1D) motion • Motion along X or Y axis • Now consider 2 dimensional motion (2D), that is between the Y and X axes • Resulting motion is a VECTOR (V) that can be resolved (broken down) into x (Vx) and y (Vy) components, as shown, where V is the size and θ the angle of the vector. Y Vy V  X Vx

  3. Geometry background • Remember the measurement baseline from geometry • Quadrants I, II, III, IV

  4. What is a vector? • A vector is a quantity that is represented by its magnitude (size/value) and its direction: • Its magnitude is represented by the length of a straight line with an arrow head showing… • its direction, represented by the angle the line makes with a reference grid (usually the + x-axis) • Example: a velocity of 20 m/s NE 

  5. Examples of vectors (vs scalars) • Displacement (vs distance) • Velocity (vs speed) • Acceleration • Force • Momentum

  6. Questions • Which of the following sentences deal with vector quantities? • I used to drive a 10-ton truck. • You’ll find a gas station if you drive 20 miles north. • I have been sailing for 20 km and still have not seen the island. • The 10-volt battery on your left is dead.

  7. What can we do with vectors? • Arithmetic add/subtract • Graphical methods • Tip-tail • Parallelogram B A C = resultant A C = resultant B

  8. What can we do with vectors? • Arithmetic add/subtracting vectors • Analytical methods • Resolve each vector (V) into its x & y components • Vx = V cos • Vy = V sin  • Then add all the x components and all the y components to get the resultant in terms of x & y • The magnitude of the resultant is “Pythagorean” • (Vx2 + Vy2) -> c2 = a2 + b2 • The direction of the resultant () is (trigonometrically) • tan-1(Vy/Vx) • Remember SOH CAH TOA

  9. 60 km 50 km x θ Practice - Navigation • A plane leaves Weyers Cave and flies north for 50km before turning east for 60km. What is the plane’s final position (x km @ θ degrees) from its start point? • Solve: 1) distance and 2) direction • 1) Pythagorean Theorem • x2 = 502 + 602 • x = 78.1 km • 2) Trigonometry • tan θ = opp/adj = 50/60 • θ = tan-1 (50/60) = 39.8º • Answer: 78.1 km @ 39.8º

  10. Additional Math tools • What if the path is NOT right angle based? • Construct right triangles and use Pythagoras • SOH CAH TOA • Use trig tools • Law of Sines • (sin A)/a = (sin B)/b = (sin C)/c • Law of Cosines • a2 = b2 + c2 - 2bc cosA • b2 = c2 + a2 – 2ca cos B • c2 = a2 + b2 – 2ab cos C

  11. Practice • A boat sails due north for 40 kms before realizing that the dog has fallen overboard, requiring a southern trip of 10 kms to retrieve the sodden canine! The boat resumed its northerly sail for 40 kms before heading west for 60 kms. • Draw the boat’s path and determine where the boat is, relative to its start point? (vector = length + direction, required)

  12. x Ө Solution • North vector = 40-10+40 = 70 km • West vector = 60 km • Resultant (using Pythagoras) • x2 = 702 + 602 = 8500 • x = 92.19 km • Angle • Ө = tan-1(opp/adj) = tan-1 (60/70) • Ө = 40.6º West of North

  13. Practice – Your Turn • A plane flies north for 50 km, west for 40 km, then SW for 20 km before sending out a “mayday” call. • Where is the plane with respect to its start point?

  14. Free Fall • Free fall

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