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1.2.1 Vectors
Definitions
• Vectors are represented on paper by arrows
o direction = _________________________________________________
o magnitude = _______________________________________________
• Examples of vectors:
o __________________________
o __________________________
o __________________________
Angular Systems
Examples
• 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?
Practice
1) Measure angle θ
(a)
θ
(b)
θ
(c)
θ
A
20 meters at 10° south of west
B
34 meters at 42° east of north
A
20 meters at ______ B
34 meters at ______ Compass Point System – uses angles
measured from a given compass direction. Uses two compass directions for a
reference.
Reference Vector System – uses due east as a 0° reference. Angles are measured from that point.
2) Add a vector to each diagram such that the angle given is between the given vector and your drawing.
3) 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!)
(a) 45°
(b) 125°
(c) 300°
(a)
A B
(b)
A
B
(c)
A
B
(d)
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(a) A
R
(b)
A R
(c)
A
R
(d) R
A
4) 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.
5) A man walks east for 6 meters, then south for 8 meters, then west for 12 meters. a. Sketch his path in the area below using a scale of 1.0 centimeter = 2.0
meters.
b. Draw the man’s final displacement vector.
c. Measure the length of the vector on your paper. ______________ cm
d. What is this measurement in meters? ______________ m
e. Use geometry to determine the magnitude of the man’s displacement.
Vector Components
• Break vector into two perpendicular components using X-Y system
• ___________________ • ___________________
– Example
– v = 6.0 m/s at 30°
• __________________________
• __________________________
• Vectors can be added together by adding their COMPONENTS
• Results are used to find:
– ____________________________
– ____________________________
30° 6.0 m/s
6) 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
7) 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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1.2.2 Relative Motion
Examples
• A train is moving east at 25 meters per second. A man on the train gets up and walks toward the front of the train at a speed of 2.0 meters per second.
o What is his velocity?
Relative to a fixed point on the train ________ m/s
Relative to a fixed point on the Earth ________ m/s
• 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 a speed of 1.5 meters per second.
o What is the passenger’s velocity?
Relative to a fixed point on the plane ________ m/s
Relative to a fixed point on the Earth ________ m/s
Review Questions
1.2.1a – Use a protractor and ruler to add vectors graphically.
- Use a protractor and ruler to construct the resultant produced when the two
vectors in each system are added together.
1.2.1b – Determine vector components.
- An object moves with a velocity of 15 meters per second in a direction that is 25° east of north.
o If due east is 0°, what is the reference angle of this vector’s velocity?
[65°]
o What is the northward component of the object’s velocty?
[6.33 m/s] o What is the eastward component of the object’s velocity?
Non-Parallel Vectors
• What happens to the overall velocity of the aircraft as the wind changes directions?
Perpendicular Kinematics
• Critical variable in multi-dimensional problems is _________________.
Practice
1) A hockey player who is 5.0 meters in front of a 1.2 meter wide goal slides puck directly at its center. The player releases the puck at 2.5 meters per second. As the puck slides toward the goal it drifts to the right at 0.4 meter per second.
a. List the givens in the chart.
b. How long does it take for the puck to reach the goal?
c. How far does it drift in this time?
d. Is it a goal?
2) Bill is sitting on a tree branch that is 4.0 meters above the ground with a bucket of water. He wants to dump the water on his friend Tim who is approaching the tree at a constant speed of 8.0 meters per second. How far should Tim be from the tree when Bill dumps the water?
Vthrust Vthrust Vthrust
Vthrust Vthrust Vthrust
vwind
vwind
vwind vwind
vwind
With no wind, the speed depends only on the
aircraft thrust…
200 m A swimmer moves across a 200 meter wide river at a velocity of
0.5 meter per second east.
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 (south) at 0.1 meter per
second. What does this mean for the swimmer?
vshot
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1.2.3 Ground Launched Projectiles
Definitions
• A projectile is an object that is moving through the air with no means of propulsion after it is launched.
o Can be launched with any ________________ at any ______________
o May be subjected to _________________________________________
Assumptions
Horizontal Vertical
vi =____________________ vi = ____________________
a = ____________________ a = ____________________
vf = ____________________
Review Questions
1.2.2a – Determine the resultant speed/velocity in a two dimensional system. - A girl rows a boat with a velocity of 2.5 meters per second due east across a
river. As she rows across the river a current of 1.5 meters per second pushes her boat due north. Sketch a set of vectors to represent these two velocities
and calculate the resultant speed of the boat. [2.9 m/s]
1.2.1b – Determine an unknown variable in a two dimensional system with constant velocity in both directions.
- A plane flies north with a speed of 100 meters per second. At the same time, a crosswind pushes the plane east with a speed of 15 meters per second. In the time that the plane flies 300 meters north, how far east will it drift?
[ 45 m]
- An ant crawls with a speed of 0.1 meter per second directly west across a conveyor belt that is 0.5 meter wide. At the same time, the conveyor belt moves south. What speed was the conveyor belt moving with if the ant moves 10 meters south in the same time that it takes it to get across the belt?
[2.0 m/s]
30° What is the horizontal part of the
soccer ball’s initial velocity? What is the vertical part of the soccer ball’s initial velocity?
range max height
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 25° above horizontal. The round leaves the gun of Tank#1 at velocity v and hits the bunker. (Ignore air resistance in this question). Explain how Tank #2 could fire from a different angle, but also hit the bunker.
4) Sketch a graph that shows the relationship between the horizontal range of a projectile and the angle at which it is launched.
5) A ball flies through a parabolic path and passes through points A, B, and C.
Compare the…
a. …vertical accelerations A, B, and C. _______________
b. …vertical velocities at A, B, and C. ________________
c. …horizontal accelerations A, B, and C. _______________
d. …horizontal velocities at A, B, and C. ________________ Range
0 45 90 Launch Angle (°)
A
B
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1.2.3d – Use equations to determine: time of flight; max height; horizontal distance; and/or launch angle for a ground-launched projectile.
- A ball is thrown with an initial velocity of 15 meters per second at an angle of 20° above the horizontal.
o Determine the vertical component of the ball’s initial velocity.
[5.13 m/s] o Determine the ball’s total time of flight.
[1.04 s]
- A rock that is projected with an initial velocity of 24 meters per second at an angle of 60° above horizontal takes 2.12 seconds to reach its maximum height.
o What is the horizontal component of the ball’s initial velocity?
[12 m/s] o What total horizontal distance will the rock travel before it lands?
[50.88 m]
- A projectile is fired with an initial vertical velocity of 12 meters per second and an initial horizontal velocity of 15 meters per second.
o Determine the maximum vertical height that the projectile will reach.
[7.3 m]
o Determine the angle with which the projectile was launched.
[39°]
1.2.3e – Relate launch angle to: time of flight; max height; and horizontal range. - Compare the time spent in the air for a rock thrown at an angle of 30° to the
time spent in the air for a rock thrown at 70°.
- Compare the max height reached by a rock thrown at an angle of 40° to the
max height for a rock thrown at 10°.
- Compare the horizontal range for a rock is thrown at an angle of 40° to the horizontal range for a rock thrown at 75°.
Review Questions
1.2.3a – Determine components of a ground-launched projectile’s initial velocity. - A projectile is launched with a velocity of 200 meters per second at an angle of
35° above the horizontal.
o What is the vertical component of the projectile’s initial velocity?
[115 m/s]
o What is the horizontal component of the projectile’s initial velocity?
[163 m/s]
1.2.3b – Explain the assumptions needed to solve a ground-launched problem.
- What can you assumer about the horizontal and vertical accelerations of
ground-launched projectiles?
[ahoriz = 0; avert = -9.81 m/s2]
- What can you assume about the relationship between the time that it takes for
a ground-launched projectile to reach its maximum height and its total time of flight?
[ttot = 2t1/2]
1.2.3c – Sketch vectors for velocity or acceleration at different positons in a projectile’s path. Compare velocities and accelerations at different points. - At each position: A, B, and C sketch a vector to represent the vertical and
horizontal velocities of the object.
- Compare the vertical speed at A to the vertical speed at B. - Compare the horizontal speed at B to the horizontal speed at C. - Compare the vertical acceleration at A to the vertical acceleration at C. - Compare the horizontal acceleration at A to the horizontal acceleration at B.
A
B V
1.2.4 Horizontal Projectiles
Over the Edge
Assumptions
• Both balls begin with no ______________________________________________
• Both fall the same __________________________________________________
• This means that the time of flight ______________________________________
Examples
• 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 3 kilometers.
– 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?
• 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?
The red ball has an initial _______________ velocity.
But does not have an initial _______________ velocity
The green ball falls from rest and has no initial velocity
________________________________ ________________________________
As the red ball rolls off the edge, a green ball is dropped from rest from
the same height at the same time A red ball rolls off the edge of a table.
What does its path look like?
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1.2.4c – Use equations to determine: time of flight; launch height; and/or horizontal distance for a horizontally-launched projectile.
- An object is thrown horizontally from a height of 10 meters with an initial velocity of 8.0 meters per second. Determine the horizontal distance that the object will travel.
[11 m]
- A ball is thrown horizontally with an initial velocity of 5.0 meters per second. The ball travels 15 meters horizontally before it lands. What vertical height was it thrown from?
[45 m]
- A projectile is fired horizontally from a height of 15.0 meters above the ground.
The projectile travels 40 meters horizontally from the point from which it was fired before it lands. What initial velocity was the projectile fired with?
[23 m/s] Review Questions
1.2.4a – Explain the assumptions needed to solve a horizontal projectile problem.
- What can you assumer about the horizontal and vertical accelerations of
horizontally launched projectiles?
[ahoriz = 0; avert = -9.81 m/s2]
- What critical assumption must you make about the initial vertical velocity of a horizontally launched projectile?
[vi = 0]
1.2.3b – Sketch vectors for velocity or acceleration at different positons in a projectile’s path. Compare velocities and accelerations at different points. - At each position: A, B, and C sketch a vector to represent the vertical and
horizontal velocities of the object.
- Compare the vertical speed at A to the vertical speed at B. - Compare the horizontal speed at B to the horizontal speed at C. - Compare the vertical acceleration at A to the vertical acceleration at C. - Compare the horizontal acceleration at A to the horizontal acceleration at B.
A B v
