Two-Dimensional Motion and Vectors Vector Operations

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)

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)

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

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°

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)

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

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

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

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.

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

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

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

Adding Vectors That Are Not Perpendicular     R Yt Xt