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Vectors

Vectors. Physics. Objectives Graphical Method Vector Addition Vector Addition Relative Velocity. Vectors and Scalars. 1.3.1 Distinguish between vector and scalar quantities, and give examples of each. method. A vector is represented in print by a bold italicized symbol, for example, F.

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Vectors

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  1. Vectors Physics Objectives Graphical Method Vector Addition Vector Addition Relative Velocity

  2. Vectors and Scalars 1.3.1 Distinguish between vector and scalar quantities, and give examples of each. method. A vector is represented in print by a bold italicized symbol, for example, F. A vector has magnitude as well as direction. Some vector quantities: displacement, velocity, force, momentum A scalar has only a magnitude. Some scalar quantities: mass, time, temperature

  3. Vector Representation 1. The length of the line represents the magnitude and the arrow indicates the direction. 2. The magnitude and direction of the vector is clearly labeled. Head Tail

  4. Scaling!!! The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn precisely to length in accordance with a chosen scale.

  5. Compass Coordinate System Direction!!! Sometimes vectors will be directed due East or due North. However we will encounter vectors in all sorts of directions and be forced to find the angle!

  6. Compass Coordinate System Use the same scale for all vector magnitudes • Δx = 30 m @ 20º E of N • V = 20 m/s @ 30º W of N • a = 10 m/s2@ 40º W of S • F = 50 N @ 10º S of E N E W To Draw direction: Ex. 20º E of N: Start w/ North and go 20° East S Navigational System?

  7. Vector Addition and Vector Resolution Vector Addition – Taking two component vectors and finding the resultant. Vector Resolution – Taking a resultant vector and finding the x and y components. Sometimes uses trig. Identities to

  8. Easy Adding…

  9. Finding Angles or Finding Sides with angle

  10. Resultant Vectors The resultant is the vector sum of two or more vectors.

  11. Vector Addition Two Steps – Find the resultant -Right Triangle 4m up 42+22 = c2 C=4.5m Find the angle = opp/adj tan 90o = tan-1(2/4) 2m left = 27o Final Answer: 4.5m 27o down right Or 4.5m 63o up left

  12. Vector Resolution • Find the x and y components of a given angle and vector (resultant) • Example: Given a vector at 60o with a magnitude 50m/s. Find the x and y velocities • Set scale. • Use trig. functions. • Find vy with sin and vx with cos • Vy=sin(60o)*50m/s • Vx =cos(60o)*50m/s

  13. Summary • Finding Magnitude of Vector (addition) • Parallel vectors: add (same direction) or subtract (opposite directions) • Perpendicular (900) – Pythagorean theorem. • Finding Magnitude of Components (resolution) • Given angle and resultant – use trig functions to find x and y components

  14. Adding Component Vectors

  15. Adding Component Vectors

  16. Adding Component Vectors

  17. Relative Velocity We already considered relative speed in one dimension; it is similar in two dimensions except that we must add and subtract velocities as vectors. Each velocity is labeled first with the object, and second with the reference frame in which it has this velocity. Therefore, vWS is the velocity of the water in the shore frame, vBS is the velocity of the boat in the shore frame, and vBW is the velocity of the boat in the water frame.

  18. Relative Velocity In this case, the relationship between the three velocities is: (3-6)

  19. Relative Velocity

  20. Relative Velocity-Sample Problem A boat is capable of traveling at 5 m/s in still water. The boat attempts to cross a river which runs at 3 m/s downstream. What is the boats velocity as it crosses the river? 5 m/s 3 m/s R

  21. Relative Velocity-Sample Problem Answer: 5.83 m/s at 30.96o downstream

  22. Relative Velocity-Sample Problem If the river is 1 km wide, how long will it take the boat to cross the river? This problem turns into a simple linear motion problem. The boat has a velocity across the river of 5 m/s and the distance straight across the river is 1 km.

  23. Relative Velocity-Sample Problem Answer: 200 seconds

  24. Relative Velocity-Sample Problem How far downstream does the boat end up? This also turns into a simple linear motion problem. Your boat is moving downstream at 3 m/s and it is in the water for 200 seconds.

  25. Relative Velocity-Sample Problem Answer: 600 m

  26. Relative Velocity-Sample Problem A railway car is moving at 2 m/s and a man starts to walk from the back of the car to the front at 1 m/s. What is his velocity relative to the ground? Answer: 3 m/s

  27. An airplane flies at an engine speed of 100 m/s at 50º W of S into a wind of 30 m/s at 200 E of N. What is the airplane’s resultant velocity? Solve using the components method!! How far has the plane traveled after 1 hr?? Answer: 75.52 m/s @ 28.54˚ S of W 168.89 miles per 1 hour

  28. You Try!!! A motor boat traveling 4.0 m/s, East encounters a current traveling 3.0 m/s, North. a. What is the resultant velocity of the motor boat? b. If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore? c. What distance downstream does the boat reach the opposite shore?

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