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Two-Dimensional Motion and Vectors

Learn how to determine resultant magnitudes and directions using Pythagorean theorem and inverse tangent function in a 2D coordinate system. Understand resolving vectors into components using sine and cosine functions.

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Two-Dimensional Motion and Vectors

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  1. Two-Dimensional Motion and Vectors Vector Operations

  2. Coordinate System in Two Dimensions • Can change the orientation of the system so that motion is along an axis • Or can apply a coordinate system along two axes • (4,-5)

  3. Determining Resultant Magnitude and Direction • Use Pythagorean theorem to find magnitude when vectors form a right triangle • (length of one leg)2 + (length of other leg)2 = (length of hypotenuse)2 • a2+b2=c2 • C equals displacement, velocity, acceleration, etc • Use inverse tangent function to find angle of the resultant • This is the direction • angle = inverse tangent of (opposite leg)/(adjacent leg) • θ=tan-1(opp/adj)

  4. Determining Resultant Magnitude and Direction • An archaeologist climbs the Great Pyramid in Giza, Egypt. The pyramid’s height is 136m and it’s width is 2.30*102m. What is the magnitude and the direction of the displacement of the archaeologist after she has climbed from the bottom of the pyramid to the top? • h = 136m w = 2.30*102m / 2 = 115m

  5. Determining Resultant Magnitude and Direction • r2 = h2 + w2 = 1362 + (115)2 = 18496 + 13225 = 31721 • r = 178m • θ = tan-1(opposite/adjacent) = tan-1(height/width) = tan-1(136/115) = 49.8°

  6. Resolving Vectors into Components • Components of a vector – the projections of a vector along the axes of a coordinate system • Use sine function to find opposite leg • opposite leg = hypotenuse * (sine of angle) • opp = hyp * (sinθ) • Use cosine function to find adjacent leg • adjacent leg = hypotenuse * (cosine of angle) • adj = hyp * (cosθ)

  7. Resolving Vectors into Components • Find the components of the velocity of a helicopter traveling 95km/h at an angle of 35° to the ground. 95km/h 95km/h 35° y 35° x

  8. Resolving Vectors into Components • opp = hyp * (sinθ) • Vertical = hypotenuse * (sinθ) • Vertical = 95(sin35) = 54 km/h • adj = hyp * (cosθ) • Horizontal = hypotenuse * (cosθ) • Horizontal = 95(cos35) = 78 km/h

  9. Adding Vectors That Are Not Perpendicular • Form right triangles from each leg of the vector • Use sine and cosine to get total x and y displacements • Use Pythagorean theorem and inverse tangent to get magnitude and direction of resultant

  10. Adding Vectors That Are Not Perpendicular • A hiker walks 27.0km from her base camp at 35° South of East. The next day, she walks 41.0km in a direction 65° North of East and discovers a forest ranger’s tower. Find the magnitude and direction of her resultant displacement between the base camp and the tower.

  11. Adding Vectors That Are Not Perpendicular R 41.0km 35 65 27.0km

  12. Adding Vectors That Are Not Perpendicular • 27km y1=hyp. cos(55) =(27)cos(55) =(-15 km) x1=hyp. sin(55) =(27)sin(55) =22km 55 y1 x1

  13. Adding Vectors That Are Not Perpendicular y2=hyp. sin(65) =(41)sin(65) =37km x2=hyp. cos(65) =(41)cos(65) =17km 41.0km Y2 65 X2

  14. Adding Vectors That Are Not Perpendicular R Yt Xt

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