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Motionin Two Dimensions

- Chapter 7.2

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.

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.

The Trajectory of a Projectile

- What does the free-body diagram look like for

force?

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?

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?

Diagram the problem

vi

Fg Fnet

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 ?

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

Perform Calculations (y)

- Now that we have , we can use the first

formula to find the final velocity. - vfy
- vy

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

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

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?

Diagram the problem

State the Known Unknown

- Known
- xi 0
- yi ___ m
- vi ___ m/s
- ? ___
- a ___ ____
- Unknown
- ymax ?
- t ?
- x ?
- viy ?
- vix ?

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

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

Perform the Calculations (x)

- x-direction
- Initial velocity
- vix
- vix
- Determine the total distance
- x
- x

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?

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.