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## Motion in Two Dimensions

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### Motion in Two Dimensions Chapter 7.2 – PowerPoint PPT presentation

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Title: Motion in Two Dimensions

1
Motionin Two Dimensions
• Chapter 7.2

2
Projectile Motion
• What is the path of a projectile as it moves
through the air?
• ?
• ?
• .
• What forces act on projectiles?
• Only , which acts only in the
y-direction.
• is ignored in projectile
motion.

3
Choosing Coordinates Strategy
• For projectile motion
• Choose the for motion where is a
factor.
• Choose the for motion. Since there are
forces acting in this direction (of course we
will neglect friction due to air resistance), the
speed will be (a __).
• Analyze motion along the y-axis from the
x-axis.
• If you solve for in one direction, you
automatically solve for in the other direction.

4
The Trajectory of a Projectile
• What does the free-body diagram look like for
force?

5
The Vectors of Projectile Motion
• What vectors exist in projectile motion?
• in both the ___ and ___ directions.
• in the ___ direction only.

vx ( )
• Why is the velocity constant in the x-direction?
• .

ax
vy ( )
• Why does the velocity increase in the
y-direction?

6
Ex. 1 Launching a Projectile Horizontally
• A cannonball is shot horizontally off a cliff
with an initial velocity of 30 m/s. If the
height of the cliff is 50 m
• How far from the base of the cliff does the
cannonball hit the ground?
• With what speed does the cannonball hit the
ground?

7
Diagram the problem
vi
Fg Fnet
8
State the Known Unknown
• Known
• xi 0
• vix ___ m/s
• yi ___ m/s
• viy ___ m/s
• a ___ ___ m/s2
• y ____ m
• Unknown
• x at y -50 m
• vf ?

9
Perform Calculations (y)
• Strategy
• Use reference table to find formulas you can use.
• vfy
• y
• Note that __ has been substituted for __ and __
for __.
• Use known factors such as in this case where the
initial velocity in the y-direction is known to
be to simplify the formulas.
• vfy
• y

10
Perform Calculations (y)
• Now that we have , we can use the first
formula to find the final velocity.
• vfy
• vy

11
Perform Calculations (x)
• Strategy
• Since you know the for the (
), you also have it for the .
• is the only variable that can transition
between motion in both the __ and __ directions.
• Since we air resistance and gravity does not
act in the (x-direction), a __.
• Choose a formula from your reference table
• x
• Since a ___, the formula reduces to x _____
• x

12
Finding the Final Velocity (vf)
• We were given the initial x-component of
velocity, and we calculated the y-component at
the moment of impact.
• Logic Since there is no acceleration in the
horizontal direction, then ____ ____
• We will use the .

vfx
vf ?
vfy
13
Ex. 2 Projectile Motion above the Horizontal
• A ball is thrown from the top of the Science Wing
with a velocity of 15 m/s at an angle of 50
degrees above the horizontal.
• What are the x and y components of the initial
velocity?
• What is the balls maximum height?
• If the height of the Science wing is 12 m, where
will the ball land?

14
Diagram the problem
15
State the Known Unknown
• Known
• xi 0
• yi ___ m
• vi ___ m/s
• ? ___
• a ___ ____
• Unknown
• ymax ?
• t ?
• x ?
• viy ?
• vix ?

16
Perform the Calculations (ymax)
• y-direction
• Initial velocity viy
• viy
• viy
• Time when vfy 0 m/s vfy viy gt (ball at
)
• t
• t
• t
• Determine the maximum height ymax
• ymax
• ymax

17
Perform the Calculations (t)
• Since the ball will accelerate due to gravity
over the distance it is falling back to the
ground, the time for this segment can be
determined as follows
• Time from peak to when ball hits the ground
• From reference table ymax
• Since ___ can be set to zero as can ___,
• t
• t
• t
• By adding the it takes the ball to reach its
maximum height ( )to the it takes to reach
the ground will give you the total .
• ttotal

18
Perform the Calculations (x)
• x-direction
• Initial velocity
• vix
• vix
• Determine the total distance
• x
• x

19
Analyzing Motion in the x and y directions
independently.
• x-direction
• dx vix t vfxt
• vix vi?cos?
• y-direction
• dy ½ (vi vf) t vavg t
• vf viy gt
• dy viy t ½ g(t)2
• vfy2 viy2 2gd
• viy vi?sin?

20
Key Ideas
• Projectile Motion
• is the only force acting on a projectile.
• Choose a coordinate axis that where the
x-direction is along the and the y-direction
is .
• Solve the x and y components .
• If is found for one dimension, it is also
known for the other dimension.