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Chapter 3: Two – Dimensional Motion and Vectors

Chapter 3: Two – Dimensional Motion and Vectors. Section 3-1 and 3-2 pages 84-97. Vectors. A scalar is a quantity that does not involve direction. 55 mph 18 cm long A vector is a quantity that involves both magnitude and direction ( velocity, acceleration, displacement, force)

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Chapter 3: Two – Dimensional Motion and Vectors

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  1. Chapter 3: Two – Dimensional Motion and Vectors Section 3-1 and 3-2 pages 84-97

  2. Vectors • A scalar is a quantity that does not involve direction. • 55 mph • 18 cm long • A vector is a quantity that involves both magnitude and direction (velocity, acceleration, displacement, force) • 55 mph north • A downward force of 3 Newtons

  3. Definition Magnitude R is represented by length Head Directionθ is represented by the angle Tail θ The resultant vector can be defined in polar coordinates as R at θN of E. Try YOURS!!

  4. Parallel Vector Addition Adding vectors in the same direction 10 23 (resultant) = 13 Adding vectors in the opposite direction 5 = 3 (resultant) 8

  5. Basic Trig Functions x R= hyp B = opp 90o θ y A = adj For the right triangle placed at the origin Sin θ = B/R = opp/hyp Cos θ = A/R = adj/hyp Tan θ = B/A = opp/adj A2 + B2 = R2

  6. Perpendicular Vector Addition For two perpendicular vectors 5 12 Construct resultant R by drawing a vector from the tail of the horizontal vector to head of the vertical vector R 5 θ = tan-1 5/12 = 22.6o R2 = 52 + 122 R = 13 θ 12

  7. Example: A boat heads east at 8.00 m/s across a river flowing north at 5.00 m/s. What is the resultant velocity of the boat? 5.00 m/s N Ө 1) Use pythagorean theory. 2) Use tan Ө = opp/hyp 8.00 m/s E R = 9.43 m/s at 32°

  8. B B R R A A D D C C E E Multiple Vector Addition Can be added in any order!! A + B + C + D + E = Distance R = Resultant = Displacement

  9. Adding Vectors SUMMARY The sum of two or more vectors is known as the RESULTANT  Vectors Acting in the Same Direction (parallel) ADD  Vectors Acting in the Opposite Direction (parallel) SUBTRACT  At 90o angles – Ah- Trigonometry. . . (perpendicular) PYTHAGOREAN TAN Ө

  10. At angles other than 90o - three methods • Graphical – scaled drawing • Resolution into Components Method – break each vector into right triangles then use trig functions • Law of Sines and Cosines c2 = a2 + b2 – 2abcosC a = b = c sin A sin B sin C

  11. Using the Graphical Method of Vector Addition: • Vectors are drawn to scale and the resultant is determined using a ruler and protractor. • Vectors are added by drawing the tail of the second vector at the head of the first (tip to tail method). • The order of addition does not matter. • The resultant is always drawn from the tail of the first to the head of the last vector. BE METICULOUS IN YOUR DRAWING!!! Your accuracy depends on it. (±2°, 0.2 cm)

  12. Method 1: Adding Vectors Graphically(It’s making a scaled drawing.) Steps: Decide what quadrant the vectors will be in. Draw the axis and write the in a box. Draw the first vector to scale starting at the origin and label it . Draw the remaining vectors, so that they make a path and label them _____, ______, _____, etc. Draw the as the dashed line from the and label it . Measure the length of ____ to get the and the angle of(from the closest axis) to get the and write your answer in a box. DIRECTION ALWAYS < 45° angle of . SCALE A TAIL TO HEAD B C D TAIL OF 1ST VECTOR TO HEAD OF LAST RESULTANT R R MAGNITUDE R DIRECTION

  13. Example: Cartman gets upset with Kenny for taking his doughnut. Cartman chases Kenny 30 meters at 40o N of E and then 20 meters at 10o E of N. Calculate Cartmans’s total displacement. Solve this graphically.

  14. Example: Cartman gets upset with Kenny for taking his doughnut. Cartman chases Kenny 30 meters at 40o N of E and then 20 meters at 10o E of N. Calculate Cartmans’s total displacement. Solve this graphically.

  15. Can add any number of vectors at once Uses simple tools No mathematical equations needed Must be correctly draw to scale and at appropriate angles Subject to human error Time consuming Advantages and Disadvantages of the Graphical Method

  16. This completes Method One! • So lets get • Vector problems #1 and #2 due tomorrow. PRACTICING!!!

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