Chapter 3 Vectors & 2-Dimensional Motion1

2 3.1 Vectors & Scalars Revisited Vector: magnitude & direction Displacement Velocity Acceleration Scalar: magnitude but no direction Temperature Speed Time intervals

Chapter 3 Vectors & 2-Dimensional Motion3 3.2 Vector Properties Vector Format Handwritten: A Printed: A, bold font Scalar Format: A, italics

Chapter 3 Vectors & 2-Dimensional Motion4 3.2 Vector Properties Vector Equality A & B are equal if they have the same magnitude & direction. Equal vectors can be moved parallel to itself without being affected

Chapter 3 Vectors & 2-Dimensional Motion5 3.2 Vector Properties Which of these vectors have the same MAGNITUDE?

Chapter 3 Vectors & 2-Dimensional Motion6 3.2 Vector Properties Which of these vectors have the same DIRECTION?

Chapter 3 Vectors & 2-Dimensional Motion7 3.2 Vector Properties Adding Vectors Must have same units Graphical Methods Triangular method of addition Parallelogram method of addition Sum is independent of order of addition  A + B = B + A  Commutative law of addition Component Method

Chapter 3 Vectors & 2-Dimensional Motion8 3.2 Vector Properties Triangle Method

Chapter 3 Vectors & 2-Dimensional Motion9 3.2 Vector Properties Parallelogram Method

Chapter 3 Vectors & 2-Dimensional Motion Vector Properties Negative of a Vector Same magnitude  opposite direction A + (-A) = 0 Subtracting Vectors A – B = A + (-B) Multiplying/Dividing by a scalar 4A, A/5

Chapter 3 Vectors & 2-Dimensional Motion Vector Properties Adding 2 Vectors Adding 3 Vectors Subtracting Vectors

Chapter 3 Vectors & 2-Dimensional Motion Vector Components V = V x + V y V x = V cos Ө V y = V sin Ө VxVx VyVy

Chapter 3 Vectors & 2-Dimensional Motion Vector Components nics/vectors/components/vectorComponents. html nics/vectors/components/vectorComponents. html

Chapter 3 Vectors & 2-Dimensional Motion14 Vector Tutorial Khan Academy Vector Tutorial Aircraft Takeoff Problem

Chapter 3 Vectors & 2-Dimensional Motion15 Practice Problems Find the x and y components of the following vectors: 240 N at 330º 34 m/s at 210º 15 m at 12º 20 m/s 2 at 90º

Chapter 3 Vectors & 2-Dimensional Motion16 Practice Problems Find the x and y components of the following vectors: 240 N at 330º F y = F x = m/s at 210º 15 m at 12º 20 m/s 2 at 90º

Chapter 3 Vectors & 2-Dimensional Motion17 Practice Problems Find the x and y components of the following vectors: 240 N at 330º F y = F x = m/s at 210º V y = 17.0 V x = m at 12º 20 m/s 2 at 90º

Chapter 3 Vectors & 2-Dimensional Motion18 Practice Problems Find the x and y components of the following vectors: 240 N at 330º F y = F x = m/s at 210º V y = 17.0 V x = m at 12º x y = 3.12 x x = m/s 2 at 90º

Chapter 3 Vectors & 2-Dimensional Motion19 Practice Problems Find the x and y components of the following vectors: 240 N at 330º F y = -120F x = m/s at 210º V y = 17.0 V x = m at 12º x y = 3.12 x x = m/s 2 at 90º a y = 20.0 a x = 0

Component Method Adding vectors using “trig” & “arithmetic” Step 1: Find all x and y components Step 2: Add up all the x components Add up all the y components Step 3:Using the “new” x and y components find the “new” resulting vector! Step 4:Sanity check

21 Vector & Projectile Motion Practice Problems Find the resultant of the following 2 vectors: 1) 100 units due west and 2) 200 units 30 o north of east.

22 Vector & Projectile Motion Practice Problems Find the resultant of the following 2 vectors: 1) 100 units due east and 2) 200 units 30 o north of east. 124 units 54 o north of west

23 Vector & Projectile Motion Practice Problems An ant on a picnic table travels 30 cm eastward, then 25 cm northward and finally 15 cm westward. What is its directional displacement with respect to its original position?

24 Vector & Projectile Motion Practice Problems An ant on a picnic table travels 30 cm eastward, then 25 cm northward and finally 15 cm westward. What is its directional displacement with respect to its original position? 59 o north of east

25 Vector & Projectile Motion Practice Problems A boy pulls a sled across a level field by exerting a force of 110 newtons at an angle of 30 o with the ground. What are the parallel and perpendicular components, respectively, of this force with respect to the ground?

26 Vector & Projectile Motion Practice Problems A boy pulls a sled across a level field by exerting a force of 110 newtons at an angle of 30 o with the ground. What are the parallel and perpendicular components, respectively, of this force with respect to the ground? 95 newtons, 55 newtons

27 Vector & Projectile Motion Practice Problems I walk 6 miles in a straight line in a direction north of east and I end up 2 miles east and several miles north. How many degrees north of east have I walked?

28 Vector & Projectile Motion Practice Problems I walk 6 miles in a straight line in a direction north of east and I end up 2 miles east and several miles north. How many degrees north of east have I walked? 71 o

Chapter 3 Vectors & 2-Dimensional Motion29 Practice Problems From the x and y components given, find the direction and magnitude of the resultant. F y = 120 N, F x = 345 N v y = 31 m/s, v x = 8 m/s

Chapter 3 Vectors & 2-Dimensional Motion30 Practice Problems A soccer ball is kicked with a horizontal velocity of 11.3 m/s and a vertical velocity of 3.5 m/s. What is the magnitude and direction of the ball's velocity? A shot putter applies a force of 415 N to a shot at an angle of 37º. What are the horizontal and vertical components of this force?

Homework Page(s) 76 & 77 #1,2,5,7,10,13,15,18,19 Due Tomorrow whether you have class or not!

Chapter 3 Vectors & 2-Dimensional Motion32 Projectile Motion

Chapter 3 Vectors & 2-Dimensional Motion33 Chapter 3 Projectile Motion

Chapter 3 Vectors & 2-Dimensional Motion34 Chapter 3 Projectile Motion Animated Projectile Motion

Chapter 3 Vectors & 2-Dimensional Motion Projectile Motion Can be described as a superposition of two independent motions in the x and y directions If air resistance is negligible, horizontal component remains constant because there is no acceleration in the horizontal direction. Vertical component is equal to the free-fall acceleration, g. Vertical component of velocity and y-direction displacement are identical to a freely falling object.

Chapter 3 Vectors & 2-Dimensional Motion Projectile Motion If you are carrying a ball and running at constant speed and wish to throw the ball so that you can catch it as it comes back down, should you (a) throw the ball at a 45 o angle above the horizontal and maintain the same speed, (b) throw the ball straight up in the air and slow down to catch it, or (c) throw the ball straight up in the air and maintain the same speed?

Chapter 3 Vectors & 2-Dimensional Motion Projectile Motion If you are carrying a ball and running at constant speed and wish to throw the ball so that you can catch it as it comes back down, should you (a) throw the ball at a 45 o angle above the horizontal and maintain the same speed, (b) throw the ball straight up in the air and slow down to catch it, or (c) throw the ball straight up in the air and maintain the same speed?

Chapter 3 Vectors & 2-Dimensional Motion Projectile Motion As a projectile moves in its parabolic path, the velocity and acceleration vectors are perpendicular to each other (a) everywhere along its path, (b) at the peak of its path, (c) nowhere along its path, or (d) not enough information is given.

Chapter 3 Vectors & 2-Dimensional Motion Projectile Motion As a projectile moves in its parabolic path, the velocity and acceleration vectors are perpendicular to each other (a) everywhere along its path, (b) at the peak of its path, (c) nowhere along its path, or (d) not enough information is given.

Chapter 3 Vectors & 2-Dimensional Motion Projectile Motion A home run is hit into the stands. The ball is hit from home plate into the center field stands along a parabolic path. What is the acceleration of the ball (a) while it is rising, (b) at the highest point of the trajectory, and (c) while it is descending after reaching the highest point? Ignore air resistance.

Chapter 3 Vectors & 2-Dimensional Motion Projectile Motion A home run is hit into the stands. The ball is hit from home plate into the center field stands along a parabolic path. What is the acceleration of the ball (a) while it is rising, (b) at the highest point of the trajectory, and (c) while it is descending after reaching the highest point? Ignore air resistance.

Chapter 3 Vectors & 2-Dimensional Motion Projectile Motion A home run is hit into the stands. The ball is hit from home plate into the center field stands along a parabolic path. What is the acceleration of the ball (a) while it is rising, (b) at the highest point of the trajectory, and (c) while it is descending after reaching the highest point? Ignore air resistance.

Chapter 3 Vectors & 2-Dimensional Motion Projectile Motion A home run is hit into the stands. The ball is hit from home plate into the center field stands along a parabolic path. What is the acceleration of the ball (a) while it is rising, (b) at the highest point of the trajectory, and (c) while it is descending after reaching the highest point? Ignore air resistance.

Could It Happen? In the movie Speed a bus traveling at nearly 68 mph is rigged with a bomb that will go off if the bus goes below 50 mph. It has to jump a 50’ gap in a bridge – could it be done? Simple explanation More involved Physics explanation

45 Vector & Projectile Motion Practice Problems A baseball thrown from the outfield is thrown from shoulder height at an initial velocity of 29.4 m/s at an initial angle of 30 o with respect to the horizontal. It is in the air for a total time interval of 3 s before it is caught by the 3 rd baseman at shoulder height level. What is the ball’s horizontal displacement?

46 Vector & Projectile Motion Practice Problems A baseball thrown from the outfield is thrown from shoulder height at an initial velocity of 29.4 m/s at an initial angle of 30 o with respect to the horizontal. It is in the air for a total time interval of 3 s before it is caught by the 3 rd baseman at shoulder height level. What is the ball’s horizontal displacement? 76.4 m

47 Vector & Projectile Motion Practice Problems A baseball thrown from the outfield is released from shoulder height at an initial velocity of 29.4 m/s at initial angle of 30 o with respect to the horizontal. What is the maximum vertical displacement that the ball reaches during its trajectory?

48 Vector & Projectile Motion Practice Problems A baseball thrown from the outfield is released from shoulder height at an initial velocity of 29.4 m/s at initial angle of 30 o with respect to the horizontal. What is the maximum vertical displacement that the ball reaches during its trajectory? 11.0 m

49 Vector & Projectile Motion Practice Problems A stone is thrown at an angle of 30 o above the horizontal from the top edge of a cliff with an initial speed of 12 m/s. A stop watch measures the stone’s trajectory time from the top of the cliff to the bottom to be 5.6 s. What is the height of the cliff?

50 Vector & Projectile Motion Practice Problems A stone is thrown at an angle of 30 o above the horizontal from the top edge of a cliff with an initial speed of 12 m/s. A stop watch measures the stone’s trajectory time from the top of the cliff to the bottom to be 5.6 s. What is the height of the cliff? 120 m

51 Vector & Projectile Motion Practice Problems A bridge that was 5 m long has been washed out by the rain several days ago. How fast must a car be going to successfully jump the stream? Although the road is level on both sides of the bridge, the road on the far side is 2 m lower than the road on this side.

52 Vector & Projectile Motion Practice Problems A bridge that was 5 m long has been washed out by the rain several days ago. How fast must a car be going to successfully jump the stream? Although the road is level on both sides of the bridge, the road on the far side is 2 m lower than the road on this side. 8 m/s

53 Vector & Projectile Motion Practice Problems A track star in the broad jump goes into the jump at 12 m/s and launches herself at 20 o above the horizontal. How long is she in the air before returning to the ground?

54 Vector & Projectile Motion Practice Problems A track star in the broad jump goes into the jump at 12 m/s and launches herself at 20 o above the horizontal. How long is she in the air before returning to the ground? 0.83 s

55 Vector & Projectile Motion Practice Problems A fireman, 50 m away from a burning building, directs a stream of water from a hose at an angle of 30 o above the horizontal. If the velocity of the stream of water is 40 m/s, at what height will the stream of water strike the building?

56 Vector & Projectile Motion Practice Problems A fireman, 50 m away from a burning building, directs a stream of water from a hose at an angle of 30 o above the horizontal. If the velocity of the stream of water is 40 m/s, at what height will the stream of water strike the building? 18.7 m