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 –a 2 +b 2 =c 2 –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*10 2 m. What is the magnitude and the direction of the displacement of the archaeologist after she ha climbed from the bottom of the pyramid to the top?  h = 136mw = 2.30*10 2 m / 2 = 115m

Determining Resultant Magnitude and Direction  r 2 = h 2 + w 2 = (115) 2 = =  r = 178m  θ = tan -1 (opposite/adjacent) = tan -1 (height/width) = tan -1 (136/115) = 49.8°

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

FFFFind the components of the velocity of a helicopter traveling 95km/h at an angle of 35° to the ground.

 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 FFFForm right triangles from each leg of the vector UUUUse sine and cosine to get total x and y displacements UUUUse Pythagorean theorem and inverse tangent to get magnitude and direction of resultant

Adding Vectors That Are Not Perpendicular