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Chapter 2: Vectors Vector: a quantity which has magnitude (how big) and direction Vectors displacement velocity acceleration force Scalars distance speed Vectors are denote by bold face or arrows V or V The magnitude of a vector is denoted by plain text V Vectors can be graphically represented by arrows direction magnitude p150c2:1 Vector Addition: Graphical Methodof R = A + B •Shift B parallel to itself until its tail is at the head of A, retaining its original length and direction. •Draw R (the resultant) from the tail of A to the head of B. B B A + = A = R p150c2:2 the order of addition of several vectors does not matter B C D A C C D A B B D A p150c2:3 If the vectors to be added are perpendicular to each other, then trigonometric methods can easily be applied opposite A sin q hypotenuse C C A q B adjacent B cosq hypotenuse C opposite A tan q adjacent B A2 B2 C 2 p150c2:4 Vector Subtraction: the negative of a vector points in the opposite direction, but retains its size (magnitude) • A- B = A +( -B) A B - = A -B R = -B A p150c2:5 Resolving a Vector replacing a vector with two or more (mutually perpendicular) vectors => components directions of components determined by coordinates or geometry. B C = A + Examples: horizontal and vertical North-South and East-West p150c2:6 Example: As a youth, Dr. Gallis walked 6 km north north east (60º north of east) to reach school. What are the East and North “components” of his displacement? N y D D y q = 60º x Dx E Example 2.3: A woman on the ground sees an airplane climbing at an angle of 35º above the horizontal. By driving at 70 mph, she is able to stay directly below the airplane. What is the speed of the airplane? y v q = 35º x vx =70 mph, horizontal p150c2:7 Vector Addition by components R=A+B+C Resolve vectors into components(Ax, Ay etc. ) Add like components Ax + Bx + Cx = Rx Ay + By + Cy = Ry The magnitude and direction of the resultant R can be determined from its components. Example 2.5: The sailboat Ardent Spirit is headed due north at a forward speed of 6.0 knots (kn). The pressure of the wind on its sails causes the boat to move sideways to the east at 0.5 kn. A tidal current is flowing to the southwest at 3.0 kn. What is the velocity of Ardent Spirit relative to the earth’s surface? p150c2:8 Motion in the vertical plane idealized horizontal motion: no acceleration idealized vertical motion: acceleration of gravity (downwards) Motion in the vertical plane is a combination of these two motions motion is resolved into horizontal and vertical components Compare motion of a dropped ball and a ball rolled off of a horizontal table v v v 0 v 0 v 0 Ax 0 Ay Bx By xB 0 x A v0t 1 2 y gt 2 v y gt p150c2:9 Example: A ball is rolled off of a 1m high horizontal table with an initial speed of 5 m/s. How long is the ball in the air? How far from the table does the ball land? What is the final velocity of the ball? What is the final speed of the ball? p150c2:10 Projectile Flight object with an initial velocity v0 at an angle of q with respect to the horizon v0 v0 x v 0 cosq v0 y v0 sin q v x v0 x v y v 0 y - gt x x 0 v0 x t q Initial velocity components Velocity components at later times 1 2 y y0 v 0 y t - gt 2 Position of projectile p150c2:11 Example 2.7: A ball is thrown at 20 m/s at an angle of 65º above the horizontal. The ball leaves the thrower’s hand at a height of 1.8 m. At what height will it strike a wall 10 m away? p150c2:12 v0 Projectile Range q R 2v0 y 1 2 y y0 v 0 y t - gt y 0 at impact t 2 g 2v0 y R x - x 0 v0 x t v 0 x g 2v 0 sin q v0 2 v0 cosq 2 sin q cosq g g v0 2 R sin 2q g R max v0 2 g q 2 90 - q 1 (angles with the same range) (at 45º ) p150c2:13 Example 2.8: An arrow leaves a bow at 30 m/s. What is its maximum range? At what two angles could the archer point the arrow for a target 70 m away? p150c2:14