Document Sample

Lecture 4: Vectors & Components Questions of Yesterday 1) A skydiver jumps out of a hovering helicopter and a few seconds later a second skydiver jumps out so they both fall along the same vertical line relative to the helicopter. 1a) Does the difference in their velocities: a) increase b) decrease c) stay the same 1b) What about the vertical distance between them? 2) I drop ball A and it hits the ground at t1. I throw ball B horizontally (v0y = 0) and it hits the ground at t2. Which is correct? a) t1 < t2 b) t1 > t2 c) t1 = t2 Vector vs. Scalar Quantities Vector Quantities: Magnitude and Direction Ex. Displacement, Velocity, Acceleration Scalar Quantities: Magnitude Ex. Speed, Distance, Time, Mass What about 2 Dimensions? Vectors in 1 Dimension 3 y (m) Direction specified 2 solely by + or - 1 x (m) x (m) -3 -2 -1 1 2 3 -1 -3 -2 -1 0 1 2 3 -2 -3 Vectors: Graphical Representation Vector Quantities: Magnitude and Direction Represent in 2D with arrow Length of arrow = vector magnitude Angle of arrow = vector direction y (m) R m at qo above x-axis 3 2 Position of vector 1 not important q q x (m) Vectors of equal -3 -2 -1 1q2 3 length & direction are equal -1 -2 Can translate vectors for q convenience (choose ref -3 frame) Adding Vectors: Head-to-Tail Must have same UNITS (true for scalars also) Must add magnitudes AND directions..how? A+B=? Head-to-Tail Method A+B Adding Vectors: Commutative Property A+B=B+A? A+B B+A YES! A+B=B+A Can add vectors in any order Subtracting Vectors A -> -A Negative of vector = 180o rotation A - B = A + (- B) A-B Multiplying & Dividing Vectors by Scalars 2 * A = 2A -2 * A = -2A Ex. v = x/t t=3s Graphical Vector Techniques N 1 box = 10 km A plane flies from base W E camp to lake A a distance 280 km at a direction 20o lake B S north of east. After dropping off supplies, the plane flies to lake B, which is 190 km and 30.0o west of north from 30o lake A. Graphically determine lake A the distance and direction from lake B to the base camp. 20o base camp Vector Components Every vector can be described by its components Component = projection of vector on x- or y- y y axis B R q y x x A R x From magnitude (R) and direction Rx = Rcosq (q) of R can determine Rx and Ry Ry = Rsinq Vector Components Can determine any vector from its components y R2 = Rx2 + Ry2 R = (Rx2 + Ry2)1/2 R q y tanq = Ry/Rx x q = tan-1(Ry/Rx) R -90 < q < 90 x Vector Components Can determine any vector Careful! from its components y R2 = Rx2 + Ry2 (-x, +y) (+x, +y) R = (Rx2 + Ry2)1/2 II I x tanq = Ry/Rx III IV q = tan-1(Ry/Rx) (-x, -y) (+x, - -90 < q < 90 y) I, IV: q = tan-1(Ry/Rx) II, III: q = tan-1(Ry/Rx) + 180o Important to know direction of vector! Vector Addition: Components Why are components useful? When is magnitude of A + B = A + B ? A B A+B R x = A x + Bx + Cx…. R = A + B + C…. = R y = A y + By + ? Cy…. q = tan-1(Ry/Rx) Vector Addition: Components lake B Using components determine the distance and direction from lake B to the base camp. 30o + Rx = Ax + Bx lake A Cx…. R y = A y + By + 20o Cy…. base camp q = tan-1(Ry/Rx) -90 < q < 90 Vector Components: Problem #2 A man pushing a mop across a floor cause the mop to undergo two displacements. The first has a magnitude of 150 cm and makes an angle of 120o with the positive x- axis. The resultant displacement has a magnitude of 140 cm and is directed at an angle of 35.0o to the positive x- axis. Find the magnitude and direction of the second displacement. Vector Components: Problem #3 An airplane starting from airport A flies 300 km east, then 350 km at 30.0o west of north, and then 150 km north to arrive finally at airport B. The next day, another plane flies directly from A to B in a straight line. a) In what direction should the pilot travel in this direct flight? b) How far will the pilot travel in the flight? Questions of the Day 1) Can a vector A have a component greater than its magnitude A? a) YES b) NO 2) What are the signs of the x- and y-components of A + B in this figure? a) (x,y) = (+,+) b) (+,-) c) (-,+) d) (-,-)

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 3 |

posted: | 8/15/2012 |

language: | English |

pages: | 17 |

OTHER DOCS BY ewghwehws

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.