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Lesson 3-1 
Review of Vectors
Scalars and Vectors 
 Recall a scalar does not have a direction 
 A vector has BOTH magnitude and direction 
 Vectors can be added graphically 
5 
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8 
9 
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4 
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5 
5 
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Similar Quantities 
 When adding vectors, the units must match 
 It would be meaningless to add a force vector to a 
velocity vector 
 They are essentially apples and oranges 
 When vectors do have the same units, we 
may add or subtract the vectors
Example 
A student is walking to school. First, the 
student walks 350m to a friend’s house. The 
two then both walk 740 m to school. 
The method to add the vectors is called the tail 
to tip method. The vector we find is called 
the resultant vector.
Moving Vectors 
 Vectors can be moved parallel to each other 
 Does not matter where the vectors are, as long as 
they are addable, tail to tip 
 Example 
 Push a toy car across a moving sidewalk 
 Say the sidewalk is moving at 1.5 m/s 
 The car is pushed .8 m/s
Vector Addition and Subtraction 
 Vector addition is commutative 
 The order the vectors are added does not matter 
 To subtract a vector, simply add the opposite
Multiplying and Dividing Vectors 
 Multiplying or Dividing vectors by scalars 
results in vectors 
 Lets say we have the velocity of a race car 
 We want to examine the properties of the car when it is 
traveling twice as fast 
 If vi is v, what is twice vi? 
 What is half of vi 
 What would be the new v if the car drove twice as fast in 
the opposite direction?
Lesson 3-2 
Vector Operations
Coordinate Systems 
 Up to this point, we have only needed one 
dimension to study our situations 
 What if we wanted to study a ball being thrown at 
45o above the ground? 
 That path of motion does not fit any of our current 
axis 
 We will have to use a combination of the two axis 
 Note: Orientation of the axis is up to you.
Determining the Resultant 
 Trigonometry is very useful to find the 
resultant vector. 
 The Pythagorean Theorem 
 Think of a tourist in Egypt walking up the side of a 
pyramid 
 Are they walking vertical? 
 Are they walking horizontal? 
 It is a combination of the two motions that produces one 
final motion, somewhere between horizontal and vertical
Resultant 
 The resultant of two vectors is also a vector 
 That means the resultant must have: 
 Magnitude 
 Direction 
 It is not enough to say the magnitude of the 
resultant, it must have direction also. 
 We will use the trig functions of sine, cosine 
or tangent to find the direction
Resolving Vectors 
 Any vector may be broken into x and y 
components 
 That is to say any vector may be RESOLVED into 
its component vectors 
 A horizontal vector has a 0 y component 
 A vertical vector has a 0 x component 
 A vector at 45o has equal x and y vectors
Examples 
 Page 92, Film Crew 
 Pg 93
Non-perpendicular Vectors 
 Until now, all of our vectors have been 
perpendicular to each other 
 Things in real life are much, much less rigid 
 Lets say a plane travels 50 km at an angle of 
35o, then levels out and climbs at 10o for 220 
km 
 These vectors are not perpendicular, we cannot 
use the Pythagorean Theorem, yet 
 Resolve the vectors
Lesson 3-3 
Projectile Motion
Two Dimensional Motion 
 In the last section, we resolved vectors into x 
and y components. 
 We will apply the same ideas to something 
thrown or flying in the air 
 Think of a long jumper 
 When she approaches her jump, she has only an 
x component 
 When she jumps, she has both x and y 
components
Analyze Projectile Motion 
 We can break the motion into the two 
component vectors and apply the kinematic 
equations one dimension at a time 
 Any object thrown or launched into the air 
that is subject to gravity is said to have 
projectile motion 
 Examples?
Projectile Path 
 Projectiles follow a path called a parabola 
 A common mistake is to assume projectiles fall 
straight down 
 Since there is vxi, there must be continuous 
horizontal motion 
Vx 
V 
Vy
Projectile Path 
Vx 
V 
Vy 
 Neglecting air resistance, is there anything to stop the 
projectile in the horizontal direction? 
 Velocity in the horizontal direction is constant 
 In real life, horizontal velocity is not constant, but for our 
purposes we will assume uniform, constant velocity
Projectile Path 
 Projectile motion is simply free fall with horizontal 
velocity 
 If two similar objects fall to the ground from the 
same height, one straight down while the other is 
thrown out to the side, which hits first? 
 It is very important to realize motion in the x 
direction is completely independent of motion in the 
y direction
Summary 
 A projectile has horizontal velocity until the 
object stops (hits the ground) 
 A projectile will have a vertical velocity that is 
ever changing due to gravity, until the 
projectile stops (hits the ground) 
 What is the only factor that is consistent in 
the x AND y directions? 
 Time
Projectile Path 
Vx 
V 
Vy 
Sample Pg 101, Practice Pg 102
Objects Launched at an Angle 
 When an object is launched at an angle, the 
object has both horizontal and vertical 
velocity components 
 This is similar to the motion of an object 
thrown straight up with an initial vertical 
velocity 
 Example pg 103, Practice pg 104
Lesson 3-4 
Relative Motion
Frames of Reference 
 Velocities are different in different frames of 
reference 
 You are in a train traveling at 40 km/h 
 Relative to the train, how fast are you moving? 
 Someone outside sees the train pass, how fast do they 
see you moving? 
 The velocities are different because the 
reference frames were different 
 You – Train, Outside observer - Earth
Examples 
 You are driving on the interstate at 80 km/h 
and a car passes you at 90 km/h 
 How fast does it seem the passing car is moving 
to you? To someone on the side of the road? 
 A semi-truck driving west at 85 km/h passes 
a car on the other side of the road, driving 
east at 75 km/h. To the trucker, how fast is 
the car moving?
Examples 
 A person standing on top of a train traveling at 20 km/h. They 
throw a baseball. How fast does it look like the ball is moving to 
a person standing on the ground when: 
 The ball is thrown 10 km/h forward 
 The ball is thrown 40 km/h backward 
 The ball is thrown 20 km/h backward 
 The ball is thrown straight up 
Example pg 108, Practice pg 109

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Bellaire High School Advanced Physics - Chapter 3 - Projectile Motion

  • 1. Lesson 3-1 Review of Vectors
  • 2. Scalars and Vectors  Recall a scalar does not have a direction  A vector has BOTH magnitude and direction  Vectors can be added graphically 5 ¾¾® ¾¾® 8 9 ¾¾® ¬¾¾ 4 17 ¬¾¾ ¬¾¾ 5 5 ¬¾¾ ¬¾¾ 9 ¾¾® 14
  • 3. Similar Quantities  When adding vectors, the units must match  It would be meaningless to add a force vector to a velocity vector  They are essentially apples and oranges  When vectors do have the same units, we may add or subtract the vectors
  • 4. Example A student is walking to school. First, the student walks 350m to a friend’s house. The two then both walk 740 m to school. The method to add the vectors is called the tail to tip method. The vector we find is called the resultant vector.
  • 5. Moving Vectors  Vectors can be moved parallel to each other  Does not matter where the vectors are, as long as they are addable, tail to tip  Example  Push a toy car across a moving sidewalk  Say the sidewalk is moving at 1.5 m/s  The car is pushed .8 m/s
  • 6. Vector Addition and Subtraction  Vector addition is commutative  The order the vectors are added does not matter  To subtract a vector, simply add the opposite
  • 7. Multiplying and Dividing Vectors  Multiplying or Dividing vectors by scalars results in vectors  Lets say we have the velocity of a race car  We want to examine the properties of the car when it is traveling twice as fast  If vi is v, what is twice vi?  What is half of vi  What would be the new v if the car drove twice as fast in the opposite direction?
  • 8. Lesson 3-2 Vector Operations
  • 9. Coordinate Systems  Up to this point, we have only needed one dimension to study our situations  What if we wanted to study a ball being thrown at 45o above the ground?  That path of motion does not fit any of our current axis  We will have to use a combination of the two axis  Note: Orientation of the axis is up to you.
  • 10. Determining the Resultant  Trigonometry is very useful to find the resultant vector.  The Pythagorean Theorem  Think of a tourist in Egypt walking up the side of a pyramid  Are they walking vertical?  Are they walking horizontal?  It is a combination of the two motions that produces one final motion, somewhere between horizontal and vertical
  • 11. Resultant  The resultant of two vectors is also a vector  That means the resultant must have:  Magnitude  Direction  It is not enough to say the magnitude of the resultant, it must have direction also.  We will use the trig functions of sine, cosine or tangent to find the direction
  • 12. Resolving Vectors  Any vector may be broken into x and y components  That is to say any vector may be RESOLVED into its component vectors  A horizontal vector has a 0 y component  A vertical vector has a 0 x component  A vector at 45o has equal x and y vectors
  • 13. Examples  Page 92, Film Crew  Pg 93
  • 14. Non-perpendicular Vectors  Until now, all of our vectors have been perpendicular to each other  Things in real life are much, much less rigid  Lets say a plane travels 50 km at an angle of 35o, then levels out and climbs at 10o for 220 km  These vectors are not perpendicular, we cannot use the Pythagorean Theorem, yet  Resolve the vectors
  • 16. Two Dimensional Motion  In the last section, we resolved vectors into x and y components.  We will apply the same ideas to something thrown or flying in the air  Think of a long jumper  When she approaches her jump, she has only an x component  When she jumps, she has both x and y components
  • 17. Analyze Projectile Motion  We can break the motion into the two component vectors and apply the kinematic equations one dimension at a time  Any object thrown or launched into the air that is subject to gravity is said to have projectile motion  Examples?
  • 18. Projectile Path  Projectiles follow a path called a parabola  A common mistake is to assume projectiles fall straight down  Since there is vxi, there must be continuous horizontal motion Vx V Vy
  • 19. Projectile Path Vx V Vy  Neglecting air resistance, is there anything to stop the projectile in the horizontal direction?  Velocity in the horizontal direction is constant  In real life, horizontal velocity is not constant, but for our purposes we will assume uniform, constant velocity
  • 20. Projectile Path  Projectile motion is simply free fall with horizontal velocity  If two similar objects fall to the ground from the same height, one straight down while the other is thrown out to the side, which hits first?  It is very important to realize motion in the x direction is completely independent of motion in the y direction
  • 21. Summary  A projectile has horizontal velocity until the object stops (hits the ground)  A projectile will have a vertical velocity that is ever changing due to gravity, until the projectile stops (hits the ground)  What is the only factor that is consistent in the x AND y directions?  Time
  • 22. Projectile Path Vx V Vy Sample Pg 101, Practice Pg 102
  • 23. Objects Launched at an Angle  When an object is launched at an angle, the object has both horizontal and vertical velocity components  This is similar to the motion of an object thrown straight up with an initial vertical velocity  Example pg 103, Practice pg 104
  • 25. Frames of Reference  Velocities are different in different frames of reference  You are in a train traveling at 40 km/h  Relative to the train, how fast are you moving?  Someone outside sees the train pass, how fast do they see you moving?  The velocities are different because the reference frames were different  You – Train, Outside observer - Earth
  • 26. Examples  You are driving on the interstate at 80 km/h and a car passes you at 90 km/h  How fast does it seem the passing car is moving to you? To someone on the side of the road?  A semi-truck driving west at 85 km/h passes a car on the other side of the road, driving east at 75 km/h. To the trucker, how fast is the car moving?
  • 27. Examples  A person standing on top of a train traveling at 20 km/h. They throw a baseball. How fast does it look like the ball is moving to a person standing on the ground when:  The ball is thrown 10 km/h forward  The ball is thrown 40 km/h backward  The ball is thrown 20 km/h backward  The ball is thrown straight up Example pg 108, Practice pg 109