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 Direction specified solely by + or - x (m) y (m) 1 2 3 -3 -2 -1 x (m) 1 2 3 -3 -2 -1

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 Position of vector not important Vectors of equal length & direction are equal Can translate vectors for convenience (choose ref frame) 2 R q R q 1 R q x (m) -3 -2 -1 1 2 3 -1 R q -2 -3

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 B A B A + B A

Adding Vectors: Commutative Property A + B = B + A ? B A B + A A + B A B YES! A + B = B + A Can add vectors in any order

Negative of vector = 180o rotation Subtracting Vectors A -> -A Negative of vector = 180o rotation A - B = A + (-B) B A A -B -B -A A - B

Multiplying & Dividing Vectors by Scalars 2 * A = 2A -2 * A = -2A A A 2A -2A x Ex. v = x/t t = 3 s v

Graphical Vector Techniques W S E 1 box = 10 km A plane flies from base camp to lake A a distance 280 km at a direction 20o north of east. After dropping off supplies, the plane flies to lake B, which is 190 km and 30.0o west of north from lake A. Graphically determine the distance and direction from lake B to the base camp. 190 km 30o lake B 20o 280 km lake A base camp

Vector Components Every vector can be described by its components Component = projection of vector on x- or y-axis Rx = Rcosq Ry = Rsinq R Rx Ry x y q From magnitude (R) and direction (q) of R can determine Rx and Ry y B = A + B R x A

Can determine any vector from its components Vector Components Can determine any vector from its components y Ry R2 = Rx2 + Ry2 R = (Rx2 + Ry2)1/2 tanq = Ry/Rx = tan-1(Ry/Rx) -90 < q < 90 R q x Rx

Can determine any vector from its components Vector Components Can determine any vector from its components Careful! y R2 = Rx2 + Ry2 R = (Rx2 + Ry2)1/2 tanq = Ry/Rx q = tan-1(Ry/Rx) -90 < q < 90 (-x, +y) (+x, +y) II I x III IV (-x, -y) (+x, -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 Rx = Ax + Bx + Cx…. Ry = Ay + By + Cy…. q = tan-1(Ry/Rx) -90 < q < 90 R = A + B + C…. = ?

Vector Addition: Components lake B Using components determine the distance and direction from lake B to the base camp. 190 km 30o Rx = Ax + Bx + Cx…. Ry = Ay + By + Cy…. q = tan-1(Ry/Rx) -90 < q < 90 lake A 280 km 20o base camp

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. In what direction should the pilot travel in this direct flight? 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? 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) (-,-) A B