VECTORS AND PROJECTILE MOTION

1. VECTORS AND PROJECTILE MOTION Chapter 3

2. SCALAR A SCALAR quantity is any quantity in physics that has MAGNITUDE ONLY Number value with units Mass, volume, density are also scalar quantities.

3. Vectors Any quantity that requires both magnitude and direction for a complete description is a vector quantity. Because direction is an important characteristic of a vector, arrows are used to represent them. The direction that the arrow is pointing represents the direction of the vector. The length of the arrow is proportional to the magnitude of the vector

4. The tip of the arrow, that is, its point, is called the head of the vector. The other end of the arrow is called the tail of the vector.

5. Examples VECTORS An arrow above the symbol illustrates a vector quantity. It indicates MAGNITUDEand DIRECTION

6. Vectors are usually drawn to scale. The green arrow is twice as long as the red arrow, indicating that it has twice the magnitude. 10 m/s 5 m/s If these were velocity vectors and the green vector represented a velocity of 10 m/s in the positive x direction, then the red vector would be interpreted as 5 m/s in the Positive x direction.

7. This vector, scaled so that 1 cm equals 20 m/s, represents a velocity of 60 m/s to the right.

8. Vector Notation In physics, a vector is usually named with a single letter with an arrow above it. In physics textbooks, the letter may be simply in a bold font with no arrow.

9. The practice of using the length of an arrow to represent the magnitude of a vector applies to any kind of vector. Vector Quantities Displacement Direction Velocity Acceleration Momentum Force Weight Drag Lift Thrust 8

10. The direction of a vector is often stated in terms of North, South, East, and West. Note:The positive or negative on an arrow only indicates its direction, such as forward (+), backward (-) or upward (+),downward (-).

11. DIRECTION N Direction using cardinal points: W of N E of N N of E N of W E W S of W S of E W of S E of S S

12. For the direction of the arrow to be meaningful, some sort of coordinate system is necessary A Cartesian coordinate system is typically used for this purpose. It consists of a pair of lines on a flat surface or plane, that intersect at right angles. 

13. (x,y) In the figure above point P1 has coordinates (3, 4), and point P2 has coordinates (-1, -3).

19. Conventions for Describing Directions of Vectors We’ve already seen that vectors can be directed due East, due West, due South, and due North. But some vectors are directed northeast (at a 45 degree angle); and some vectors are even directed northeast, yet more north than east. Thus, there is a clear need for some form of a convention for identifying the direction of a vector that is not due East, due West, due South, or due North.

20. The vector has a magnitude of 2 km and a direction 30° North of East. If the car in the diagram had moved 4 km instead of 2 km, the arrow would have been drawn twice as long.

21. In cases where the direction of a vector that is not due East, due West, due South, or due North, the direction of a vector can be represented as degrees. 11 The Physics Classroom

23. The direction of a vector can be expressed as a counterclockwise angle of rotation of the vector about its “tail” from due East. • Using this convention, a vector with a direction of 40 degrees is a vector that has been rotated 40 degrees in a counterclockwise direction relative to due east.

24. A vector with a direction of 240 degrees is a vector that has been rotated 240 degrees in a counterclockwise direction relative to due east.

25. Representing the Magnitude of a Vector The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn a precise length in accordance with a chosen scale. For example, the diagram below shows a vector with a magnitude of 20 miles. Since the scale used for constructing the diagram is 1 cm = 5 miles, the vector arrow is drawn with a length of 4 cm. That is, 4 cm x (5 miles/1 cm) = 20 miles.

26. Based on the scale, what length would you draw the vector shown above? 5 cm

27. Question? During a relay race, runner A runs a given distance due north and then hands off the baton to runner B, who runs for the same distance in a south-easterly direction. Are they equal? No, they are not equal! In order for two vectors to be equal, the magnitude and direction of both vectors must be the same.

28. Equivalent Vectors These two vectors are equivalent. They have the same length and direction. You can move vectors around on the coordinate system. So long as you do not change their length or orientation they are equivalent.

29. A vector having the same magnitude but opposite direction to a vector A, is -A. These two vectors are not equal. Even though their magnitudes appear to be the same, their directions are not the same.

30. (a) The displacement vector for a woman climbing 1.2 m up a ladder is represented by the letter D. (b) The displacement vector for a woman climbing 1.2 m down a ladder is represented by the letter -D

31. The force vector for a man pushing on a car with 450 • N of force in a direction due east isF. • The force vector for a man pushing a car with 450 • N of force in a direction due west is –F. Note:The positive or negative on an arrow only indicates its direction, such as forward (+), backward (-) or upward (+),downward (-). Forces are neither positive or negative.

32. P and Q are two vectors represented in the diagram below. P and Q are represented as two vectors having equal magnitudes but opposite directions therefore, |P| = |Q|.

33. ADDING VECTORS ADDITION: When two (2) vectors point in the SAMEdirection (collinear), simply add them together. EXAMPLE: A man walks 46.5 m east, then another 20 m east. Calculate his displacement relative to where he started. + 20 m, E 46.5 m, E MAGNITUDE relates to the size of the arrow and DIRECTION relates to the way the arrow is drawn 66.5 m, E

35. VECTOR SUBTRACTION SUBTRACTION: When two (2) vectors point in the OPPOSITE direction, simply subtract them. EXAMPLE: A man walks 46.5 m east, then another 20 m west. Calculate his displacement relative to where he started. 46.5 m, E - 20 m, W 26.5 m, E

37. I walk 100 m north and then 200 m south. What is my total displacement from my starting point? 100 m N – 200 m S = -100 m South

38. Examples Note: The positive or negative on an arrow only indicates its direction, such as forward (+), backward (-) or upward (+),downward (-). Vectors are neither positive nor negative. The Physics Classroom

39. Addition of Vectors – Graphical Methods • Head to tail method • The tail of one vector is placed at the head on the other vector. • Neither the direction or length of either vector is changed. • A third vector is drawn connecting the tail of the first vector to the head of the second vector. • This third vector is called the resultant vector. • Measure its length to find the magnitude then measure its direction to fully describe the resultant

40. Head To Tail rule of vector addition In the diagram below, two perpendicular vectors are represented by two sides of a right triangle in sequence. In sequence means that the vectors are placed such that the tail of vector Cbegins at the arrow head of the vector placed before it, B. The third closing side of the triangle A = B + C, represents the sum (or resultant) of the two vectors in both magnitude and direction.

41. In this case, the closing side of the right triangle A represents the sum (i.e. resultant) of individual displacements Band C. Draw the resultant vector by connecting the tail of the first vector to the head of the last vector.. A = B + C

42. The direction of the resultant vector must be included. The vector’s direction is show as an angle, measured from the horizontal. If drawn to scale, a ruler is used to measure the magnitude of the resultant vector and a protractor is used to measure the angle  (direction) of the resultant vector.

43. The head to tail rule does not restrict to perpendicular vectors Also, you can start with any vector. In figure (i), the ruleis applied starting with vector b. In figure (ii) the ruleis applied starting with vector a. In either case, the resultant vector, c, is same in magnitude and direction.

44. NON-COLLINEAR VECTORS When two (2) vectors are PERPENDICULAR to each other, you must use the PYTHAGOREAN THEOREM Example: A man travels 120 km east then 160 km north. Calculate his resultant displacement. FINISH the hypotenuse is called the RESULTANT 160 km, N VERTICAL COMPONENT START 120 km, E HORIZONTAL COMPONENT

45. DIRECTION – ANGLE! Just putting N of E is not good enough (how far north of east ?). We need to find a numeric value for the direction. To find the value of the angle we use a Trig function called TANGENT. 200 km 160 km, N q = 53.1o N of E 120 km, E So the COMPLETE final answer is : 200 km, 53.1 degrees North of East

46. Example A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north. 8.0 m/s, W 15 m/s, N V q The Final Answer : 17 m/s, @ 28.1 degrees West of North

47. Head To Tail The head to tail method can be applied to any number of vectors (also called polygon method).

48. Addition of Vectors-Sample Problem • A hiker walks 2 km to the North, 3 km to the West, 4 km to the South, 5 km to the East, 1 more km to the South, and finally 2 km to the West. How far did he end up from where he started? Hint: What is his resultant? Shown is his path, notice all of the vectors are head to tail The resultant is in Red. 3 km, South

49. Velocity Vectors Speed is a measure of “how fast”; velocity is a measure of both how fast and “which direction. The 60-km/h crosswind blows the 80-km/h aircraft off course at 100 km/h. Pythagorean Theorem: Resultant = √[(60km/h)2 + (80km/h)2]= 100 km/h Direction:

50. Adding Vectors Therefore, adding vectors that act along parallel directions is intuitive: if they are in the same direction, they add; if they are in opposite directions, they subtract. The sum of two or more vectors is called their resultant. To find the resultant of two vectors that don't act in exactly the same or opposite direction, we use the parallelogram rule. Construct a parallelogram wherein the two vectors are adjacent sides—the diagonal of the parallelogram shows the resultant. In the figure bellow the parallelograms are rectangles.