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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.
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