Ch.3 Scalars & Vectors

Transcription

1 Ch.3 Scalars & Vectors Scalar: e.g. Vector: e.g. Vector Notation: using vector A. A or A (text books bold) (writing on paper) On paper, vectors are represented as with magnitude (size) and direction. 25m/s E 250m 45 o 1

2 Adding Vectors Vectors can only be added if they are in the e.g. velocity + velocity, acc. + acc. but NOT velocity + acc. Vectors are added via the Method or the method. When adding vectors, the overall result is known as the Vector. (R) i.e. R = A + B aka: Nett vector The Triangle Method In this method, the vector arrows are added. e.g. Diagrammatically show R = A + B where A & B are: A 40m/s Need to discuss scale: So 40m/s could be drawn and the 20m/s would be B 20m/s 45 o cm long. cm 45 o 2

3 - Solve graphically. (Use a BLOODY ruler! And a protractor) Measure the length of R and the angle. Using the scale, convert the length to a magnitude. So R is o above the horizontal axis. Solve mathematically using: Law of Cosines & Law of Sines. R 20m/s 135 o 45 o 40m/s c 2 = a 2 + b 2 2abCos 3

4 SinB = SinC b c The Parallelogram Method In this method, the vector arrows are added, and a parallelogram is formed. Use the previous example and solve again. Note: R = A + B is the same as R = B + A So it doesn t matter which arrow you start off with. 4

5 Referencing Direction When stating the direction, you need to include an e.g. B 45 o 20m/s Vector B is Or Another option: Bearings i.e. N, E, W, S etc E.g. A plane is flying from LA to Miami at a speed of 180.0m/s (~400mph) on an Easterly bearing. It encounters a crosswind of 45.00m/s directly South. What will be the plane s true speed and direction? Solve graphically & mathematically m/s East R 45.00m/s South 5

6 Negative Vectors A negative vector has the same magnitude as the positive vector, but points in the direction. A -A B 45 o 6

7 Subtracting Vectors Vectors can only be added. To subtract, you must. i.e. to solve: you must do: R = A - B R = A + (-B) E.g. Solve R = A + B & R = A - B where: A 40m/s B 20m/s 45 o R = A + B A R 135 o B R = A B = A + (-B) 7

8 Multiplication/Division of Vectors Multiplication/division of a vector increases/decreases the only of a vector by a given factor. (a scalar number). The is NOT affected. E.g. 3B is 3 times the magnitude of B but still in the same direction. 15m/s 45m/s B 3B So: Scalar x Vector = Vector (diff. mag, same dir.) e.g. time x velocity = displacement What about Vector x Vector =? Vector Components A vector can be represented by its x & y components. E.g. A 5m 30 o Can be represented by: 5m A 30o So A = 8

9 We can now say: Cosθ = A x A Sinθ = A y A Tanθ = A 2 = Projectile Motion When an object is launched into the air, it is solely under the influence of. If this projectile is launched at an angle, it will follow a path in an shape. This is known as Projectile Motion. θ Assumptions: i.) v x is constant because a x = 0 (no drag) ii.) a y = -g = -9.80m/s 2 iii.) v y = the highest pt (Apex, Zenith ) 9

10 What happens to each of the components during the flight? v yo v o θ v xo v xo = v yo = 10

11 Projectile Motion Equations x-direction v xo = v o.cosθ x = v xo.t y-direction v yo = v o.sinθ v yf = v yo gt y = v yo.t ½ gt 2 v yf2 = v yo2 2gy Question: Which will take longer to hit the ground: A ball that rolls off the edge of a table, or a ball that is launched horizontally off a table? Answer: They both fall at the same rate and hit the ground at the same time. 11

12 Example: A bird with a worm in it s beak is flying horizontally at 5m/s at a height of 25m. The worm wriggles itself free. From the moment of release, (a) how long will it take for the worm to hit the ground, and (b) how far along the ground will it land? A fireworks rocket is launched at an angle of 25 o to the ground. If it takes off at a speed of 12.0 m / s, then: a.) How long till it lands, b.) How far away will it land, and c.) What maximum height will it reach? 12

13 Example: Robbie Knievel the DareDevil is attempting another stunt. He will ride his motorcycle at 30m/s up a 32 o angle ramp that reaches a height of 20m. (a) What is the maximum height a wall can be for him to still clear it, (b) The total time in the air (c) How far along will he land, and (d) What velocity will he land at? 30m/s 32 o 20m x v o = 30m/s = 32 o h = 20m a.) h max =? b.) t Total =? c.) x = d.) v f =? v xo = = = v yo = = = 13

14 v xf v yf v f 14

15 Challenge question: What angle will achieve the greatest distance in the x direction? Ans: = o (Could you prove it?) Note: If A + B = o, then x from angle A = x from angle B 15