Download Vectors [1

REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Vectors
Representing Vectors
• Vectors on paper are simply arrows that show:
• DIRECTION represented by the way _______________________________________________
• MAGNITUDE represented by the _________________________________________________
• Examples of vectors:
_______________________________________________________________________________
Vector Diagrams
1. A _____________________ is clearly listed
2. A vector _________________ (with arrowhead) is drawn in a specified direction. The vector
arrow has a __________ and a _____________.
3. The _________________________ of the vector is clearly labeled.
4. Vectors can be moved ____________________ to themselves in a diagram without changing the
value.
Direction of vectors
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Changing Systems
• What is the reference vector angle for a vector that points 50 degrees east of south?
• What is the reference vector angle for a vector that points 20 degrees north of east?
Things we can DO with vectors
• Add/Subtract with a vector – to produce _____________________________
• Multiply/Divide by a vector or a scalar to produce either a _____________________ or
_______________________
PROTRACTOR PRACTICE
1) Measure angle θ.
2) Add a vector to each diagram such that the angle given is between the given vector and your drawing.
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Vector Addition
•
The ______________________ is the vector sum of two or more vectors. It is the result of
adding two or more vectors together
Two methods for adding vectors
•
Graphical method: using a scaled vector diagram
•
–
_____________________________________
–
_____________________________________
Mathematical method - ______________________ theorem and ________________________
methods
Vector addition: head-to-tail method
+
A
+
=
B
=
C
(Resultant)
Parallelogram method
+
A
+
=
B
=
C
(Resultant)
Vector addition practice
1) For each of the sets of vectors below (A and B), sketch what you think the resultant sum of the two
vectors will look like. (Hint: imagine each arrow as a person pulling with a force proportional to the
arrow length!)
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
2) Below are groups of resultant vectors (R) paired with a single component (A). Sketch the missing
vector that when added to vector A would produce the resultant.
Steps for adding vectors using head and tail method
1. Choose a _________ and indicate it on a sheet of paper. The best choice of scale is one that will
result in a diagram that is as large as possible, yet fits on the sheet of paper.
2. Pick a starting location and draw the first vector to scale in the indicated direction. Label the
magnitude and direction of the scale on the diagram (e.g., SCALE: 1 cm = 20 m).
3. Starting from where the head of the first vector ________, draw the second vector to scale in the
indicated direction. Label the magnitude and direction of this vector on the diagram.
4. Repeat steps 2 and 3 for all vectors that are to be added
5. Draw the _____________ from the tail of the first vector to the head of the last vector. Label this
vector as Resultant or simply R.
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
6. Using a ruler, measure the length of the resultant and determine its magnitude by converting to
real units using the scale (4.4 cm x 20 m/1 cm = 88 m).
7. Measure the direction of the resultant using the reference vector convention.
Practice – draw a scaled diagram

A man walks east for 3 meters, then south for 5 meters, then west for 6 meters.
a. Draw his path in the area below using a scale of 1 centimeter = 1 meter.
b. Draw the man’s final displacement vector.
c. Measure the length of the vector on your paper: _________________.
d. Calculate the man’s final displacement in meters: ________________.
e. Devise a way to solve this problem using your knowledge of geometry. Explain your method and
show your work.
f. How do the results of the two methods compare to one another?
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
The commutative property of vectors
A + B = __________
Vector subtraction
A
-
=
B
?
= A
=
+
(- B )
+
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Comparing vector addition and vector subtraction
A
B
Examples
•
1.
2.
3.
4.
•
1.
2.
3.
4.
A 5.0-newton force and a 7.0-newton force act concurrently on a point. As the angle
between the forces is increased from 0° to 180°, the magnitude of the resultant of the two
forces changes from
0.0 N to 12.0 N
2.0 N to 12.0 N
12.0 N to 2.0 N
12.0 N to 0.0 N
A 3-newton force and a 4-newton force are acting concurrently on a point. Which force
could not produce equilibrium with these two forces?
1N
7N
9N
4N
•
As the angle between two concurrent forces decreases, the magnitude of the force
required to produce equilibrium
1. decreases
2. increases
3. remains the same
Vector subtraction
A
-
=
B
?
= A
=
+
(- B )
+
vector addition vs. subtraction
A
B
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Example
•
1.
2.
3.
4.
•
1.
2.
3.
4.
A 5.0-newton force and a 7.0-newton force act concurrently on a point. As the angle
between the forces is increased from 0° to 180°, the magnitude of the resultant of the two
forces changes from
0.0 N to 12.0 N
2.0 N to 12.0 N
12.0 N to 2.0 N
12.0 N to 0.0 N
A 3-newton force and a 4-newton force are acting concurrently on a point. Which force
could not produce equilibrium with these two forces?
1N
7N
9N
4N
•
As the angle between two concurrent forces decreases, the magnitude of the force
required to produce equilibrium
1. decreases
2. increases
3. remains the same
Add vectors mathematically
The procedure is restricted to the addition of two vectors
that make _______________________ to each other.
•
Using tangent function to determine a Vector's Direction
Hyp.
opp.
opp.
tanθ = adj.
θ
adj.
θ = tan-1
•
( opp.
adj. )
Example: Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east.
Determine Eric's resulting displacement.
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Example
• An archaeologist climbs the Great Pyramid in Giza, Egypt. If the pyramid’s height is 136 m
and its width is 2.30 x 102 m, what is the magnitude and the direction of the
archaeologist’s displacement while climbing from the bottom of the pyramid to the top?
Vector Components
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
EXAMPLE
1) Calculate the x and y components of the following vectors.
a. A = 7 meters at 14°
b. B = 15 meters per second at 115°
c. C = 17.5 meters per second2 at 276°
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Adding Vectors Algebraically
•
Vectors can be added together by adding their
______________________________________
•
Results are used to find
– RESULTANT MAGNITUDE
– RESULTANT DIRECTION
Example
Add vectors D and F by following the steps below.
a. Calculate the components of vectors D and F.
D = 35 meters at 25° F = 55 meters at 190°
b. Calculate the sum of the x-components of vectors D and F.
c. Calculate the sum of the y-components of vectors D and F.
d. Sketch the resultant x and y vectors on the axes below.
e. Calculate the length of the resultant generated by the resultant components
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Example
A bus heads 6.00 km east, then 3.5 km north, then 1.50 km at 45o south of west. What is the total
displacement?
A: 6.0 km, 0° CCW
B: 3.5 km, 90° CCW
C: 1.5 km, 225° CCW
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Relative Motion
ALL motion is RELATIVE!
Examples
A train is moving east at 25 meters per second. A man on the train gets up and walks toward the front at
a speed of 2 meters per second.
How fast is he going? - Depends on what we want to relate his speed to!!!
Relative to a fixed point on the train
_____________________
_____________________
A passenger on a 747 that is traveling east at 230 meters per second walks toward the lavatory at the
rear of the airplane at 1.5 meters per second.
How fast is the passenger moving? - Again, depends on how you look at it!
_____________________
____________________
Non-Parallel Vectors
What happens to the aircraft’s speed as the wind changes direction?
Perpendicular Kinematics
• Critical variable in multi dimensional problems is ___________________
• We must consider each dimension SEPARATELY using TIME as the only crossover variable
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Example
A swimmer moves at 0.5 meters per second across a 200 meter wide river. How long will it take the
swimmer to get across?
Now, assume that as the swimmer moves ACROSS the river, a current pushes him DOWNSTREAM at
0.1 meter per second.
The time to cross is unaffected! The swimmer still arrives on the other bank in 400 seconds. What
IS different?
PRACTICE
1) A hockey player who is 5.0 meters in front of a goal slides a puck directly at the center of a 1.2 meter
wide goal. The player shoves the puck at 2.5 meters per second. As it slides toward the goal the puck
drifts to the right at 0.4 meters per second.
a. List the givens.
b. How long does it take for the puck to reach the goal?
c. How far does it drift in this time?
d. Will the puck go into the goal or will it go wide?
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
2) A motorboat traveling 4 m/s, East encounters a current traveling 3.0 m/s, North.
a. What is the resultant velocity of the motorboat?
b. If the width of the river is 80 meters wide, then how much time does it take the boat to travel
shore to shore?
c.
What distance downstream does the boat reach the opposite shore?
3) Bill is sitting on a tree limb that is 4 meters above the ground with a bucket of water. He wants to
dump the water so that it falls on his friend Tim who is approaching on a bike at a constant speed of 8
meters per second. How far away should Tim be from a point directly under Bill when Bill dumps the
water?
This is more challenging, but its not THAT hard! You need to separate your system into horizontal
and vertical parts. You also need to realize that in the horizontal part, Tim moves at a constant speed,
but the water dropped by Bill will accelerate. Other than that, everything else is the same and time is
again the crossover variable
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Ground Launched Projectiles
What is a projectile?
• An object that is thrown into the air with some ___________________
• Can be launched at any _________________
• In _____________________ after launch (no outside forces except force of gravity)
•
The path of the projectile is a ________________________________
What is the vertical part of the soccer ball’s initial velocity? _________________________________
What is the horizontal part of the soccer ball’s initial velocity? ________________________________
Pythagorean Theorem: ____________________________________________
What do we know or assume about the vertical part of a projectile problem?
• Initial vertical velocity = __________________
• Acceleration = _________________
• Vertical speed will be _________at the maximum height
• Time to top = ______________ total time in the air
– Find time to top using final velocity equation:
____________________________________________________

Vertical Distance – ______________________
– Use time to top and solve vertical distance equation
__________________________________________
What do we know or can we assume about the horizontal part of a projectile problem?
• Initial horizontal velocity = __________________
• Acceleration = _________________ (if we assume no air resistance)
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
• Horizontal Distance – ______________________
– Use total time and solve horizontal distance equation _______________________________
The symmetrical nature of a ground launched projectile
Effect of air resistance
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Summary:
•
A projectile is any object upon which the only force is _______________________
•
Projectiles travel with a ____________________ trajectory due to the influence of gravity,
•
In horizontal direction: ax = _______________, its velocity is constant: vx = _____________
•
In vertical direction,: ay = ____________________. its velocity of a projectile changes by -9.81 m/s each
second,
Example
The horizontal motion of a projectile is independent
of its vertical motion.
•
•
An object was projected horizontally from a tall cliff. The diagram represents the path of the
object, neglecting friction. Comparing the following at point A & B:
1) Acceleration
2) Horizontal velocity
3) Vertical velocity
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
•
A projectile is fired with initial horizontal velocity at 10.00 m/s, and vertical velocity at 19.62 m/s.
Determine the horizontal and vertical velocity at 1 – 5 seconds after the projectile is fired.
Time
Horizontal Velocity
Vertical Velocity
0s
10 m/s, right
+19.62 m/s
1s
2s
3s
4s
5s
Equation for projectile motion
General Kinematics Equations
d = ½ (vi + vf)t
vf = vi + at
d = vit + ½at2
vf2 = vi2 + 2ad
Vertical
Horizontal
ttot = 2t1/2
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Examples
•
Practice A: A water balloon is launched with a speed of 40 m/s at an angle of 60 degrees to the
horizontal.
•
Practice B: A motorcycle stunt person traveling 70 mi/hr jumps off a ramp parallel to the
horizontal.
•
Practice C: A springboard diver jumps with a velocity of 10 m/s at an angle of 80 degrees to the
horizontal.
•
A machine fired several projectiles at the same angle, θ, above the horizontal. Each projectile
was fired with a different initial velocity, vi. The graph below represents the relationship between
the magnitude of the initial vertical velocity, viy, and the magnitude of the corresponding initial
velocity, vi, of these projectiles. Calculate the magnitude of the initial horizontal velocity of the
projectile, vix, when the magnitude of its initial velocity vi, was 40. meters per second.
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
•
A football is kicked with an initial velocity of 25 m/s at an angle of 45-degrees with the horizontal.
Determine the time of flight, the horizontal displacement, and the peak height of the football.
•
A long jumper leaves the ground with an initial velocity of 12 m/s at an angle of 28-degrees above the
horizontal. Determine the time of flight, the horizontal distance, and the peak height of the longjumper.
•
A cannon elevated at an angle of 35° to the horizontal fires a cannonball, which travels the path
shown in the diagram. [Neglect air resistance and assume the ball lands at the same height above
the ground from which it was launched.] If the ball lands 7.0 × 102 meters from the cannon 7.0
seconds after it was fired,
1. What is the horizontal component of its initial velocity?
2. What is the vertical component of its initial velocity?
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
PRACTICE
1) A baseball is hit with an initial velocity of 35 meters per second at an angle of 18° above horizontal.
a. Sketch the baseball’s trajectory. Indicate the point at which the ball’s vertical velocity is 0.
b. Calculate the vertical component of the ball’s initial velocity.
c. Calculate the time that it will take for the ball to reach the top of its flight. Explain how this amount of
time is related to the total amount of time that the ball will remain in the air.
d. Calculate the maximum height that the ball will reach.
2) A place-kicker strikes a football with a horizontal velocity of 22 meters per second and vertical velocity
of 10 meters per second. The ball reaches its maximum height within 1.02 seconds.
a. What is the total time that the ball will spend in the air? ___________
b. Calculate the horizontal distance that the football will travel during its flight.
3) Two tanks attack an enemy bunker on opposite sides from equal distance. Tank #1 fires its cannon at
the bunker with an angle of 24° above horizontal. The round leaves the gun of Tank#1 at 110 meter per
second and hits the bunker. (Ignore air resistance in this question)
Tank #2 fires a shot at a completely different angle from Tank #1, but also hits the bunker. Explain how
this is possible. You should be able to give two possible explanations.
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Horizontal Projectiles
Over the Edge
A red ball rolls off the edge of a table,
As the red ball rolls off the edge, a green ball is
dropped from rest from the same height at the
same time
What does its path look like?
Which ball hits the ground first? _________________________________________________________
Look at the Components
The red ball has an initial
The green ball falls from rest and
has no initial velocity
__________________________
_______________ velocity.
But does not have an initial
__________________________
_______________ velocity
____________
One Dimension at a Time
• Both balls begin with no _______________________
• Both fall the same ____________________________
• Find time of flight by solving in either vertical or horizontal dimension.
• We can find an object’s displacement in either dimension using ________________________________
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
Examples
• A bullet is fired horizontally from a gun that is 1.7 meters above the ground with a velocity of 55 meters
per second. At the same time that the bullet is fired, the shooter drops an identical bullet from the same
height.
–
Which bullet hits the ground first?
• An airplane making a supply drop to troops behind enemy lines is flying with a speed of 300 meters per
second at an altitude of 300 meters.
–
How far from the drop zone should the aircraft drop the supplies?
• A stuntman jumps off the edge of a 45 meter tall building to an air mattress that has been placed on the
street below at 15 meters from the edge of the building.
-
What minimum initial velocity does he need in order to make it onto the air mattress?
• A CSI detective investigating an accident scene finds a car that has flown off the edge of a cliff. The car
is 79 meters from the edge of the 25 meter high cliff.
– What was the car’s initial horizontal velocity as it went off the edge?
•
The path of a stunt car driven horizontally off a cliff is represented in the diagram below. After leaving
the cliff, the car falls freely to point A in 0.50 second and to point B in 1.00 second.
1. Determine the magnitude of the horizontal component of the velocity of the car at point B.
[Neglect friction.]
2. Determine the magnitude of the vertical velocity of the car at point A.
3. Calculate the magnitude of the vertical displacement, dy, of the car from point A to point B.
[Neglect friction.]
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REGENTS PHYSICS – CHAPTER 2 – KINEMATICS /2-D GUIDED NOTES
PRACTICE
1) A matchbox car is rolled off the edge of a 0.9 meter high table. The toy flies off the edge with a
horizontal speed of 2 meters per second. (Neglect air resistance in this problem)
a. List the givens for this problem. Remember our assumptions for horizontal projectiles!
b. Calculate the car’s time of flight. (You should be working in the vertical dimension)
c. Calculate the distance away from the table with which the car will land. (You should be working in the
horizontal dimension)
2) An astronaut on the Moon throws a wrench horizontally with a speed of 0.5 meters per second from a
height of 1.5 meters. The astronaut simultaneously drops a feather from the same height. Which object
will hit the ground first, the hammer or the feather? Why?
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