CHAPTER THREE

1. CHAPTER THREE VECTORS AND TWO-DIMENSIONAL MOTION

2. Vectors and Scalars Each of the physical quantities that we shall encounter in physics can be categorized as either a vector quantity or a scalar quantity. A vector is a physical quantity that requires both the specification of both direction and magnitude (size). Examples: Displacement, velocity, acceleration. A scalar is a quantity that can be completely specified by its magnitude with appropriate units; it has no direction. Examples: Temperature, mass, time, number of pages in your physics book.

3. Properties of Vectors When a vector is handwritten, it is often represented with an arrow over the letter. Vectors can also be represented graphically by arrows.

4. Properties of Vectors When representing vectors graphically, two things must be done first: You must choose a scale to draw your vector. Example: 1 cm = 10 m/s, 1 inch = 5 km The arrow (vector) must be pointing in the direction of the quantity. Example: North, south, east, west, etc.

5. Properties of Vectors Example: Arrows have been drawn representing the velocity of a car a various places as it rounds a curve. The green arrows represent the velocity vector at each position

6. Properties of Vectors Examples: A woman is driving her car at 88 km/hr due east. Represent this quantity graphically. A man walks 45 meters to the northeast. Represent this quantity graphically. Superman flies off the top of a building at 20 m/s at an angle of 35? below the positive x-axis. Represent this quantity graphically.

7. Properties of Vectors A very important technique in the analysis of many physical situations is the addition (and subtraction) of vectors. By adding or combining such quantities (vector addition), you can obtain the overall, or net, effect that occurs � the resultant (vector sum). Because vectors are quantities that have directions as well as magnitude, they must be added in a special way. We will use simple arithmetic for adding vectors if they are in the same direction.

8. Properties of Vectors Examples: If a person walks 8 km east one day, and 6 km east the next day, how far has the person displaced himself? If a person walks 8 km east on the first day, and 6 km west on the second day, how far has the person displaced himself?

9. Properties of Vectors Simple arithmetic cannot be used if the two vectors are not along the same line. Example: Suppose a person walks 10.0 km east and then walks 5.0 km north. What is the total displacement of the person? These displacements can be represented on a graph in which the positive y-axis points north and the positive x-axis points east. On this graph we draw an arrow to represent the displacement vector of the 10.0 km to the east, then we draw a second arrow to represent the 5.0 km displacement to the north. After taking this walk, the person is now 10.0 km east and 5.0 km north of the point of origin. The resultant displacement is represented by another arrow, starting at the origin and ending at the head of the second arrow. Using a ruler and a protractor, you can measure on this diagram that the person is 11.2 km from the origin at an angle of 27? with the positive x-axis. The magnitude can also be obtained using the theorem of Pythagoras. A2 + B2 = C2 You can only use the Pythagorean theorem when the vectors are perpendicular to each other.

10. Properties of Vectors General rules for graphically adding two vectors together, no matter what angles they make, to get their sum: On a diagram, draw one of the vectors (to scale) starting at the origin. Next draw the second vector (to scale), placing its tail at the tip of the first vector and being sure that its direction is correct. The arrow drawn from the tail of the first vector to the tip of the second represents the sum, or resultant, of the two vectors. The length of the resultant can be measured with a ruler and compared to the scale. Angles can be measured with a protractor. This method is known as the tail-to-tip method of adding vectors. NOTE: It is not important in which order the vectors are added. The tail-to-tip method can be extended to three or more vectors.

11. Properties of Vectors Examples: Consider a football player who runs 18 yards to the north, then turns and runs 10 yards to the west. Draw arrows representing his displacement and find his total displacement. Consider a student walking to school. The student walks 350 m east to a friend�s house, then 740 m north to the school. Draw arrows representing the students displacement and find the total displacement.

12. Properties of Vectors To subtract a vector, add its opposite. Given a vector V, we define the negative of this vector (-V) to be a vector with the same magnitude as V but opposite in direction. NOTE: No vector is ever negative in the sense of its magnitude: the magnitude of every vector is positive. A minus sign tells us about its direction. The difference between two vectors is equal to the sum of the first plus the negative of the second. Our rules for addition of vectors can be applied using the tail-to-tip method.

13. Properties of Vectors A second way to add two vectors is the Parallelogram method. In this method, the two vectors are drawn starting from a common origin, and a parallelogram is constructed using these two vectors as adjacent sides. The resultant is the diagonal drawn from the common origin. It is a common error to draw the sum vector as the diagonal running between the tips of the two vectors, but this is incorrect.

14. Properties of Vectors Examples: A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction to the next town. She then drives in a direction 60.0? south of east for 47.0 km to another town. What is her displacement from the post office. (Use the parallelogram method to find your answer.) An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km; the second leg is southeast (45?) for 440 km; and the third leg is at 53? south of west, for 550 km. What is the plane�s total displacement? (Use the parallelogram method to find your answer.)

15. Components of a Vector Probably the most widely used analytical method for adding multiple vectors is the component method. Adding vectors graphically using a ruler and a protractor is often not sufficiently accurate and is not useful for vectors in three dimensions.

16. Components of a Vector Consider a vector (V) that lies in a particular plane. It can be expressed as the sum of two other vectors, called components of the original vector. The components are usually chosen to be along two perpendicular directions. The process of finding the components is known as resolving the vector into its components.

17. Components of a Vector This vector V is resolved into its x and y components by drawing dashed lines from the tip (A) of the vector and drawing these lines perpendicular to the x and y axes. Then the lines OB and OC represent the x and y components of V. These vector components are written Vx and Vy.

18. Components of a Vector The scalar components, Vx and Vy, are numbers, with units, that are given a positive or negative sign depending on whether they point along the positive or negative x or y axis. The use of trigonometric functions can help find the components of a vector. Vy = Vsin? Vx = Vcos? ? is chosen to be the angle that the vector makes with the positive x-axis.

19. Components of a Vector These components form two sides of a right triangle, the hypotenuse of which has the magnitude. The resultants magnitude and direction are related to its components through the Pythagorean theorem and the definition of the tangent: Vx2 + Vy2 = V2 tan? = Vy/Vx To solve for the angle ?, we can re-write the above tangent equation to say: = tan-1 (|Vy/Vx|) When finding the angle, make sure that your calculator is set to calculate angles measured in degrees not radians.

20. Components of a Vector Example: Find the horizontal and vertical components of the 100-m displacement of a superhero who flies from the top of a tall building along the path shown in the figure to the right.

21. Components of a Vector Vectors can be conveniently written in terms of unit vectors. A unit vector is defined to have a magnitude exactly equal to one (1). It is useful to define unit vectors that point along coordinate axes, and in a rectangular coordinate system these unit vectors are called i (x), j (y), and k (z).

22. Components of a Vector Sometimes you may see unit vectors written with a �hat.� Any vector (V) can be written in terms of components as: V = Vxi + Vyj + Vzk We will only be solving for vectors in two dimensions and not three, so we will not be using the z-axis (k). Example: A ball�s displacement from the origin could be written: d = (4.50 m)i + (12.6 m)j. Where i and j are unit vectors in the x and y directions. Unit vectors are helpful when adding vectors analytically by components.

23. Components of a Vector Procedures for Adding Vectors by the Component Method Resolve the vectors to be added into their x and y components. Use the acute angles (angles less than 90?) between the vectors and the x-axis, and indicate the directions of the components by plus (+) and minus (-) signs. Add all of the x-components together, and all of the y-components together vectorially to obtain the x and y components of the resultant, or vector sum. V = Vxi + Vyj = (Vx1 + Vx2 + �)i + (Vy1 + Vy2 + �)j Express the resultant vector, using: The component form � for example, V = Vxi + Vyj The magnitude-angle form.

24. Components of a Vector Example: Let�s apply the procedural steps of the component method to the addition of the vectors in the following diagram.

25. Components of a Vector Example: A hiker begins a trip by first walking 25.0 km southeast from her base camp. On the second day she walks 40.0 km in a direction 60.0? north of east, at which point she discovers a forest ranger�s tower. Determine the components of the hiker�s displacements on the first and second days. Determine the components of the hiker�s total displacement for the trip.

26. Components of a Vector Example: An airplane is traveling 635 km/hr in a direction 41.5? west of north. Find the components of the velocity vector in the northerly and westerly directions. How far north and how far west has the plane traveled after 3.00 hours?

27. Projectile Motion A projectile is any object that is thrown or launched into the air and is subject to gravity. Anyone who has observed a baseball in motion has observed projectile motion. The ball moves in a curved path, and its motion is simple to analyze if we make two assumptions: The free-fall acceleration (gravity) is constant over the range of motion and is directed downward. The effect of air resistance is negligible. With these assumptions, we find that the path of a projectile (trajectory) is always a parabola.

28. Projectile Motion Galileo was the first to accurately describe projectile motion. He showed that it could be understood by analyzing the horizontal and vertical components of the motion separately. For convenience, we will assume that the motion begins at t = 0 at the origin of an xy coordinate system (xi and yi = 0).

29. Projectile Motion Example: Imagine a ball fired horizontally. There is no initial component of the velocity in the vertical direction (viy); the ball sails straight off horizontally at a constant velocity equal to its initial velocity (vix). On Earth, objects fall freely downward with a constant acceleration (gravity). So, the projectile will continuously descend faster and faster, as it progresses laterally, sweeping out a smooth arc that curves increasingly downward.

30. Projectile Motion Question? Suppose a bullet is fired horizontally from a pistol, and simultaneously another bullet is dropped from the same height. Which bullet will hit the ground first? They will both hit at the same time. The time it takes to reach the ground is determined solely by the vertical motion of the object. Although the two bullets have different horizontal velocities, their vertical motions are identical. Assuming that gravity is the only force acting on the bullets, the acceleration (g) of each bullet must be in the vertical direction.

31. Projectile Motion Projectile motion is just free-fall with an initial horizontal velocity. So, we can combine the kinematic equations from Chapter 2 to analyze the vertical motion of a falling body. Projectile Motion Equations: x = xi + vixt y = yi + viyt � 1/2gt2 vy = viy � gt vx = vix

32. Projectile Motion Problem Solving Strategy w/ Projectile Motion Choose an origin and an xy coordinate system. Make a sketch of the path of the projectile. Analyze the horizontal (x) motion and vertical motion (y) separately. If you are given the initial velocity, you may want to resolve it into its x and y components. List the known and unknown quantities (g = 9.81 m/s2). Remember that vx never changes throughout the trajectory, and that vy = 0 at the highest point of any trajectory that returns downward. The velocity just before hitting the ground is generally not zero. Think for a minute before jumping into the equations. A little planning goes a long way. Apply the relevant equations. You may need to combine components of a vector to get the magnitude and direction. You will usually have to solve for time, even if it is not asked for. If the object is launched horizontally, viy = 0.

33. Projectile Motion Example: A youngster hurls a ball horizontally at a velocity of +10.0 m/s from a bridge 50.0 m above a river. How long will it take for the ball to hit the water below? What is the velocity (magnitude and direction) of the ball just before it hits the water? How far (horizontally) from the bridge will it land?

34. Projectile Motion Example: An Alaskan rescue plane drops a package of emergency rations to stranded hikers. The plane is traveling horizontally at +40.0 m/s at a height of 100.0 m above the ground. Where does the package strike the ground relative to the point at which it was released? What are the horizontal and vertical components of the velocity of the package just before it hits the ground? Find the velocity (magnitude and direction) of the package just before it hits the ground.

35. Projectile Motion Example: A long jumper leaves the ground at an angle of 20.0� to the horizontal and at a velocity of 11.0 m/s. How far does he jump? What is the maximum height reached by the jumper?

36. Projectile Motion Example: A stone is thrown upward from the top of a building at an angle of 30.0� to the horizontal with an initial velocity of 20.0 m/s. The height of the building is 45.0 m. How long is the stone �in flight?� What is the final velocity of the stone just before it strikes the ground? Where does the stone strike the ground?