Projectile Motion: Solving Problems
With Angles
Ch. 5 in your text book
Students will be able to:
1) Calculate the horizontal and vertical velocity components of a velocity vector
2) Solve projectile motion problems involving angles
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Projectile Motion: Solving Problems With Angles, Exercises of Mathematical Physics
1) Calculate the horizontal and vertical velocity components of a velocity vector. 2) Solve projectile motion problems involving angles ...
Typology: Exercises
2022/2023
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Projectile Motion: Solving Problems
With AnglesCh. 5 in your text book
Students will be able to:1)^ Calculate the horizontal and vertical velocity components of a velocity vector2)^ Solve projectile motion problems involving angles
Velocity Vector Components
We saw this diagram previously showing the horizontal and verticalcomponents to a velocity vector.
v^ v^ H
v^ v
θ
Usually, you are told the velocity of an object and the angle at whichit was initially moving. To answer questions, you’ll need to find thehorizontal and vertical components to that velocity.
Velocity Vector Components
Pull out your calculators and make sure you are in “degree” mode.
sinሺߠሻ ൌ
݄ݕ^
݄݆݀ܽൌ ሻߠcosሺ
ݕ
Example: Find sin(30) and cos(30).^ sinሺ30ሻ ൌ
1 2
cos ሺ30ሻ ൌ
(^3 )
Velocity Vector Components
Each side of the triangle represents something about theobject’s velocity. Hyp.^ Adj.
Opp.
θ sinሺߠሻ ൌ
݄ݕ^
݄݆݀ܽൌ ሻߠcosሺ
ݕ θ is the variable for an angle
v^ v^ H
vv
θ sinሺߠሻ ൌ
ݒ௩ ݒ^
cosሺߠሻ ൌ
ݒ ݒ
Velocity Vector Components
Example: An object is kicked at a velocity of 2 m/s at an angle of 30°.Find the horizontal and vertical velocity components.
2 m/s30° v^ H
= 1 m/svv = 1.73 m/s
*make sure you are in “degree” mode
ݒൌ vsinሺߠሻ௩^
ݒ
ൌ vcosሺߠሻ
ݒൌ ሺ2ሻsinሺ30ሻ ݒ௩^
ൌ ሺ2ሻcosሺ30ሻ
ݒൌ^ ௩^
2 ሺ.5ሻ
ݒൌ ሺ2ሻ ሺ.866ሻ^
Velocity Vector Components
Check for understanding: An football is punted with an initial velocityof 12 m/s at an angle of 65°. Find the horizontal and vertical velocitycomponents.
12 m/s
65°vH
= 10.88 m/svv = 5.07 m/s
*make sure you are in “degree” mode
ݒൌ vsinሺߠሻ௩^
ݒൌ vcosሺߠሻ^
ൌ ሺ12ሻsinሺ65ሻ
ൌ ሺ12ሻcosሺ65ሻ
ൌ 10.88 ݉ /ݏ
ൌ 5.07 ݉ /ݏ
Solving Problems
Keep the vertical information separate from the horizontal information.Vertical Information
Horizontal Informationv = 42.44 m/s
v^ = 24.5 m/si^
x =?
v= -24.5 m/sf^ a = -10 m/s
2 x = 0
t =?
t =?
ݒ ൌ ݔ
ݐ ܽ ݐ ଶ 2 0 ൌ 24.5 ݐ
ሺെ10ሻሺݐሻ
ଶ 2
ݐൌ 4.9 ݏ vf= at + v^
i -24.5 = -10t + 24.
t = 4.9 s
Additional Angle Information
When something is shot/kicked/thrown horizontally, the angle is 0°.
v = 10 m/s
0° vH
vv= 0 m/s^ = 10 m/s
*make sure you are in “degree” mode
ݒൌ vsinሺߠሻ௩^
ݒൌ vcosሺߠሻ^
ൌ ሺ10ሻsinሺ0ሻ
ൌ ሺ10ሻcosሺ0ሻ
ൌ 0 ݉ /ݏ
ൌ 10 ݉ /ݏ
Solving Problems
Key concepts to remember:At it’s highest point, the
vertical
velocity of an object is 0
For the
vertical
direction, the acceleration will always be due to gravity (-9.81 m/s
2 on Earth) When something is dropped, it’s initial
vertical
velocity is 0
When something is thrown
horizontally
, it’s initial
vertical
velocity is 0
Solving Problems
Key concepts to remember:Velocities and displacements are vectors; they can be positive or negativeThe initial and final velocities for something that goes up and down will bein opposite directions but have the same magnitude.The time it takes an object to go up and down is twice as long at it takes togo up.The horizontal velocity won’t change while the object is in flight.
Solving Problems
Example: A ball is shot horizontally at 3 m/s and is in the air for 3seconds. How far did it go?What key words are there? Are they asking about horizontal or vertical information?
x = vtx = (3)(3)x = 9m
Solving Problems
Example: A ball is shot horizontally at 3 m/s and is in the air for 3seconds. How far did the ball fall in that time?What key words are there? Are they asking about horizontal or vertical information?
ݒ ൌ ݔ
ݐ ܽ ݐ ଶ 2 ݔൌ 0
ሺെ10ሻሺ3ሻ
ଶ (^2) ݔൌ െ45݉
Given
vertical
information: v^ = 0i^ a = -10 m/s
2 t = 3 s
Solving Problems
Example: A ball is shot from the ground at an angle. The verticalvelocity was initially 5 m/s. How long did it take before it came backto the ground?What key words are there? Are they asking about horizontal or vertical information?