2 Dimensional Vectors

Transcription

1 2 Dimensional Vectors

2 Vectors that are not collinear must be added using trigonometry or graphically (with scale diagrams) Vector quantities are drawn as arrows, the length of the arrow indicates the magnitude and the direction of the arrow corresponds to the vector direction

3 In vector diagrams the greater the magnitude, the longer the arrow The sum of 2 or more vectors is called the resultant vector

4 Vectors are added by placing the tail of one vector at the head of the other vector, the resultant is drawn from the tail of the first vector to the head of the second Resultant vector Vector 2 Vector 1

5 Vector directions There are 2 common methods of stating vector directions The direction of the resultant must be given in the same way as used in the question

6 Angles in standard position are measured counterclockwise from + x axis 90 o 180 o 0 or 360 o 270 o

7 Vector 2 32 o Vector 1 Vector 1: 54 o Vector 2: 122 o Vector 3: 292 o 54 o 22 o Vector 3

8 Compass method uses angles from north, south, east or west

9 Vector 2 N Vector 1 Vector 1: 64 o N of E or 26 o E of N 46 o Vector 2: 46 o W of N W 64 o E Or 44 o N of W Vector 3: 22 o E of S 22 o Vector 3 Or 68 o S of E S

10 Adding Vectors Can use scale diagrams Need ruler and a protractor Draw a labelled vector diagram

11 Example A person walks 700 m east and then 1.50 km south. Determine the resultant displacement graphically. (Use a scale of 1 cm = 100 m)

12 Draw axis Draw an arrow pointing east 7 cm long (700 m) N Draw an arrow pointing south (15 cm long) 700 m E Resultant is drawn from tail of 1 st vector to head of last vector Direction is measured between tail of 1 st vector and tail of resultant S 1.50 km

13 Measure resultant: multiply length by the scale factor: Line should be close to 16.6 cm which would equal 1.66 km Measure angle with protractor: should be close to 65 o

14 Resultant vector: 1.66 km at 65 o S of E

15 Example Marci walks 13 m parallel to the +x axis and then 8.5 m parallel to the y axis. Determine her displacement using a scale diagram. Answer: 15.5 m at 326 o

16 An object has 2 forces acting on it, p and q. Vector p has a magnitude of 3.1 N and vector q has a magnitude of 3.5 N. Determine the resultant force (1.0 cm = 1.0 N) Example

17 Solution Draw x-y axis with the object at the origin. Move vector q to the head of vector p Vector q is 4 squares in the y direction and +1 in the positive x direction

18 Draw resultant vector from object to the head of vector q. Solution Measure angle from x- axis Measure length of resultant, multiply by the scale. Answer: 3.9 N at 332 o

19 Example Mike walks 100 m west, then 150 m south and then 50 m east. Determine his displacement using a scale diagram.

20 Example Carla runs 100 m west then 150 m at 30 o S of W. Determine her displacement.

21 Adding vectors mathematically More accurate than scale diagrams Easily used when adding more than 2 vectors Need a labeled sketch VECTOR diagram

22 Practice Determine the angle cos cos adjacent hypotenuse 34.0m 45.9m m cos 45.9m 34.0 m 45.9 m 42.2

23 Practice Determine the angle tan tan opposite adjacent 68.9m 74.0m m tan 74.0m 68.9 m 74.0 m 43.0

24 Example A person walks 1.0 km at 000 and then turns and walks 2.0 km at 90 o. Find the resulting displacement (magnitude and direction) of the person.

25 Sketch a labeled vector diagram Use Pythagoras theorem to determine the hypotenuse d 1.0 km 2.0 km Use trig to determine the angle

26 d (1.0km) 2 (2.0km) 2 tan opposite adjacent d d 5.0km km tan tan 2.0 km 1.0 km o

27 Example A person walks 1.43 km west and then turns and walks 2.12 km south. Determine the magnitude and direction of the resultant.

28 N d (1.43 km) 2 (2.12 km) 2 W 2.12 km 1.43 km d S d d km km tan opposite adjacent tan 2.12 km 1.43 km tan ref 56.0 displacement = 2.56 km at 56.0 o S of W

29 Example Becky walks 1.50 km towards 90.0 o, turns walks 2.00 km towards 180 o, then 2.80 km towards 270 o. Determine her resulting displacement. Draw a sketch vector diagram

30 2.00 km 1.50 km Add all vectors in x direction d x = -2.0 km 2.80 km Add all vectors in y-direction d d y = 1.5 km km = km

31 Sketch a vector addition diagram Use trig to determine the resultant 2.0 km 1.3 km d

32 d = 2.39 km inside vector diagram = o In standard position = 213 o

33 Example The starship Enterprise is moving at 2.98 x 10 7 m/s when Space Cadet (5th Class) Curt pushes the wrong engine controls. As a result, Enterprise was given velocity vectors of 1.50 x 10 6 m/s at right angles to its initial velocity and 2.00 x 10 6 m/s in the opposite direction to the initial velocity. Oh no!! Not again

34 a. Determine the resulting velocity of the Starship, relative to its original direction of travel. v 3 v result v 2 Original direction v 1

35 b. Scotty gets the ship under control. Sulu reports that Starbase 13 is 6.00 x m from Enterprise on its new course and velocity. Determine the time in days required to reach the Starbase where Space Cadet (8th Class) Curt will spend the rest of his Starfleet career in charge of mops and buckets.

36 Answers v = 2.78 x 10 7 m/s at 3.09 o to its original course t = seconds t = 2.49 days

37 P 90 #5, 6 Practice