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TWO-DIMENSIONAL MOTION AND VECTORS

TWO-DIMENSIONAL MOTION AND VECTORS. I. Vectors and Scalar Quantities. Scalar Quantities  Completely described by magnitude only, and have no direction. Scalar quantities include mass, volume, time *Examples 1. speed ---- 30 m/s 2. distance --- 30 m 3. age --- 30 years

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TWO-DIMENSIONAL MOTION AND VECTORS

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  1. TWO-DIMENSIONAL MOTION AND VECTORS

  2. I. Vectors and Scalar Quantities Scalar Quantities Completely described by magnitude only, and have no direction. Scalar quantities include mass, volume, time *Examples 1. speed ---- 30 m/s 2. distance --- 30 m 3. age --- 30 years 4. time – 3:00 ………………..

  3. Vector Quantities A quantity that requires both a magnitude and direction Velocity is a vector quantity because it includes a direction, and so is acceleration. *Examples: 1. velocity --- 30 m/s north 2. displacement – 30 m left 3. acceleration – 30 m/s2 up 4. force – 30 newtons south

  4. Can be drawn as an arrow on a graph 1. the length of the arrow shows the magnitude (drawn to a scale) of the vector 2. the angle of the arrow shows the direction of the vector [ θ head tail

  5. 3. x,y axis N Negative X, Positive Y Positive X, Positive Y (x,y) (-,+) (x,y) (+,+) W E (x,y) (-,-) (x,y) (+,-) Negative X, Negative Y Positive X, Negative Y S

  6. II. Vectors can be added in different ways A. Parallelogram method (not very accurate but it gives you an idea of what the resultant vector –R-will look like) • draw the 2 vectors to scale with their TAILS touching • draw a parallel copy of each vector to make a rectangle • draw the diagonal from the point the 2 tails are touching – this line is the resultant –R- of the 2 vectors added.

  7. Resultant: A vector representing the sum of two or more vectors

  8. Adding and subtracting vectors – Same Direction If the vectors are equal in direction, add the quantities to each other. Example: 3 m east 14 m east 11 m east the resulting vector is

  9. Adding and subtracting vectors – Opposite Directions If the vectors are exactly opposite in direction, subtract the quantities from each other. Example: 3 m west 8 m east 11 m east the resulting vector is

  10. B.Graphical vector addition • draw the vectors to scale place head to tail (you can move the vectors around on the graph as long as you don’t change the length or direction)

  11. Draw an arrow connecting the other two parts of the vector & this arrow is the resultant of -R- adding the 2 vectors • the vectors that were added to get R are called components • x-component runs parallel to the x-axis • y-component runs parallel to the y-axis

  12. Components: The projections of a vector along the axes of a coordinate system. The components can be either positive or negative numbers with units. • Vectors have two parts (components) • X component – along the x axis • Y component – along the y axis

  13. To find components • To find components, you must use trigonometric functions Hypotenuse Opposite ø Adjacent

  14. III. Trig. method of vector addition A. pros: 1. great because you don’t have to draw everything to scale! 2. more accurate than graphical 3. more practical in the real world B. cons: 1. must have a right triangle 2. must use trig functions sine, cosine & tangent (sin, cos, tan)

  15. C. Right Triangle info Pythagorean Theorem--- a2 + b2 = c2 OR x2 + y2 = R2

  16. Hypotenuse = resultant vectorc c/R hypotenuse b/y opposite a/x adjacent

  17. sin θ = y/R cos θ = x/R tan θ = y/x θ = sin-1 (y/R) θ = cos-1 (x/R) θ = tan-1 (y/x) the –1 means the inverse button on your calculator

  18. Example A hiker leaves camp and hikes 11 km, north and then hikes 11 km east. Determine the resulting displacement of the hiker.

  19. Steps for finding the components • Draw a picture (arrowheads, original vector & components) • Choose a trig function • Use algebra to solve for the desired variable & plug in • Calculator in degrees! • Check with Pythagorean theorem

  20. Example

  21. X component • cos Θ = _adjacent_ hypotenuse • cos 35 = _adjacent_ 316 • 316 cos 35 = adjacent • 259 N = adjacent

  22. Y component • sin Θ = _opposite_ hypotenuse • sin 35 = _opposite_ 316 • 316 sin 35 = opposite • 181 N = opposite

  23. How to find components when you add two vectors • Find the x and y component for both vectors • Add up the x components • Add up the y components • Draw a new set of vectors • Use Pythagorean theorem to get the magnitude of the resultant vector • Use arctangent to get the angle of the new vector

  24. X component adj = hyp cos Θ adj = 36 cos34º adj = + 29.8 m Y component opp = hyp sin Θ opp = 36 sin34º opp = +20.1 m Vector d1

  25. X component opp = hyp sin Ø opp = 23 sin64º opp = - 20.7 m Y component adj = hyp cos Θ adj = 23 cos64º adj = +10.1 m Vector d2

  26. Total X displacement – add d1 and d2 dtotal = d1 + d2 dtotal = 29.8 m + (-20.7m) dtotal = +9.1m

  27. Total Y displacement – add d1 and d2 dtotal = d1 + d2 dtotal = 20.1 m + 10.1m dtotal = +30.2m

  28. To get the magnitude of the resultant vector • Use Pythagorean Theorem dTotal = (dX)2+ (dy)2 dTotal = (9.1)2+ (30.2)2 dTotal = 82.81+ 912.04 dTotal = 994.85 = 31.5 m

  29. To find the angle of the resultant vector • Use arctangent function: Θ = tan-1 (opp/adj) Θ = tan-1 (30.2/9.1) Θ = tan-1 (3.3) Θ = 73.1°

  30. Calculatinga resultant vector If two vectors have known magnitudes and you also know the measurement of the angle (θ) between them, we use the following equation to find the resultant vector. R2 = A2 + B2 – 2ABcosθ Use this for angles other than 90º Make sure your calculator is set to DEGREES! (go to MODE)

  31. Example 1: θ = 110° R2 = 5.02 + 4.02 – 2(5.0)(4.0)(cos 110) R2 = 54.68 R = 7.39 N, Southwest 4.0 N, SW R 5.0 N, W θ

  32. R2 = 4.32 + 5.12 – (2)(4.3)(5.1)(cos 35) R2 = 8.57 R = 2.93 m, northwest R 4.3 m θ 5.1 m Example 2: θ = 35º

  33. D. to add vectors: • write the givens • make a sketch • use x2 + y2 = R2to get magnitude (no neg. numbers)

  34. use trig functions to get θ (the angle/direction) θ = tan-1 (y/x) • if you just have R & θ you can get the x-component & the y-component a. find the x & y component (part) of the vector given θ = cos-1 (x/R)----- cos θ (R) = x θ = sin-1 (y/R)------ sin θ (R) = y

  35. E. If you don’t have a right triangle: • find the x & y component (part) of each vector given θ = cos-1 (x/R)----- cos θ (R) = x θ = sin-1 (y/R)------ sin θ (R) = y • add all the x-components to get total x • add all the y-components to get total y

  36. (watch for components going west or south...they will be negative & when you add the x components up this will matter) • use the Pythagorean theorem to get magnitude of resultant x2 + y2 = R2 • use trig function to get angle θ = tan-1 (y/x)

  37. Formulas • a2 + b2 = c2 • R2 = a2 + b2 - 2ab(cosθ) • SOH • CAH • TOA

  38. IV. Projectile Motion Components of Vectors--Any vector can be broken down into a horizontal and vertical component. • Projectile Motion: Free Fall with an initial horizontal velocity.

  39. A. A projectile is any object that follows motion in a curved path and continues in motion with only gravity acting on it. B. The trajectory is the path followed by a projectile. It depends on 2 INDEPENDENT vectors (have both magnitude & direction). 1. made of constant velocity horizontal motion x-component 2. constant acceleration vertical motion (gravity) y-component

  40. C. The horizontal and vertical components are completely independent... velocity has no effect on gravity! 1. Neglecting friction, the horizontal component has constant velocity. Much like a ball rolling across a table.     2. Neglecting friction/air resistance, the vertical component follows the same rules as an object in free-fall. Gravity only acts in the vertical direction it doesn’t act sideways!

  41. D. The path of a projectile is called a parabola (like McDonald’s arches). Example: -Cannonball shot from cannon -Spacecraft circling Earth -Ball thrown into the air

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