Introduction to 2D Motion

1
Introduction to 2D Motion
2
2-Dimensional Motion

• Definition motion that occurs with both x and y
  components.
• Example
• Playing pool .
• Throwing a ball to another person.
• Each dimension of the motion can obey different
  equations of motion.

3
Solving 2-D Problems

• Resolve all vectors into components
• x-component
• Y-component
• Work the problem as two one-dimensional problems.
• Each dimension can obey different equations of
  motion.
• Re-combine the results for the two components at
  the end of the problem.

4
Sample Problem

• You run in a straight line at a speed of 5.0 m/s
  in a direction that is 40o south of west.
• How far west have you traveled in 2.5 minutes?
• How far south have you traveled in 2.5 minutes?

5
Sample Problem

• A roller coaster rolls down a 20o incline with an
  acceleration of 5.0 m/s2.
• How far horizontally has the coaster traveled in
  10 seconds?
• How far vertically has the coaster traveled in 10
  seconds?

6
Sample Problem

• A particle passes through the origin with a speed
  of 6.2 m/s in the positive y direction. If the
  particle accelerates in the negative x direction
  at 4.4 m/s2
• What are the x and y positions at 5.0 seconds?
• What are the x and y components of velocity at
  this time?

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Projectile Motion

• Something is fired, thrown, shot, or hurled near
  the earths surface.
• Horizontal velocity is constant.
• Vertical velocity is accelerated.
• Air resistance is ignored.

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1-Dimensional Projectile

• Definition A projectile that moves in a vertical
  direction only, subject to acceleration by
  gravity.
• Examples
• Drop something off a cliff.
• Throw something straight up and catch it.
• You calculate vertical motion only.
• The motion has no horizontal component.

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2-Dimensional Projectile

• Definition A projectile that moves both
  horizontally and vertically, subject to
  acceleration by gravity in vertical direction.
• Examples
• Throw a softball to someone else.
• Fire a cannon horizontally off a cliff.
• Shoot a monkey with a blowgun.
• You calculate vertical and horizontal motion.

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Demo

• Monkey gun

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Horizontal Component of Velocity
Newton's 1st Law

• Is constant
• Not accelerated
• Not influence by gravity
• Follows equation
• x Vo,xt

12
Horizontal Component of Velocity
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Vertical Component of Velocity
Newton's 2nd Law

• Undergoes accelerated motion
• Accelerated by gravity (9.8 m/s2 down)
• Vy Vo,y - gt
• y yo Vo,yt - 1/2gt2
• Vy2 Vo,y2 - 2g(y yo)

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Horizontal and Vertical
15
Horizontal and Vertical
16
Launch angle

• Definition The angle at which a projectile is
  launched.
• The launch angle determines what the trajectory
  of the projectile will be.
• Launch angles can range from -90o (throwing
  something straight down) to 90o (throwing
  something straight up) and everything in between.

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Zero Launch angle

• A zero launch angle implies a perfectly
  horizontal launch.

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Sample Problem

• The Zambezi River flows over Victoria Falls in
  Africa. The falls are approximately 108 m high.
  If the river is flowing horizontally at 3.6 m/s
  just before going over the falls, what is the
  speed of the water when it hits the bottom?
  Assume the water is in freefall as it drops.

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Sample Problem

• An astronaut on the planet Zircon tosses a rock
  horizontally with a speed of 6.75 m/s. The rock
  falls a distance of 1.20 m and lands a horizontal
  distance of 8.95 m from the astronaut. What is
  the acceleration due to gravity on Zircon?

20
Sample Problem

• Playing shortstop, you throw a ball horizontally
  to the second baseman with a speed of 22 m/s. The
  ball is caught by the second baseman 0.45 s
  later.
• How far were you from the second baseman?
• What is the distance of the vertical drop?

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Today Zero Launch Angle

• Demonstration.
• Homework Questions???

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General launch angle

• The situation is more complicated when the launch
  angle is not straight up or down (90o or 90o),
  or perfectly horizontal (0o).

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General launch angle

• You must begin problems like this by resolving
  the velocity vector into its components.

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Resolving the velocity

• Use speed and the launch angle to find horizontal
  and vertical velocity components

Vo
?
25
Resolving the velocity

• Then proceed to work problems just like you did
  with the zero launch angle problems.

Vo
?
26
Sample problem

• A soccer ball is kicked with a speed of 9.50 m/s
  at an angle of 25o above the horizontal. If the
  ball lands at the same level from which is was
  kicked, how long was it in the air?

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Sample problem

• Snowballs are thrown with a speed of 13 m/s from
  a roof 7.0 m above the ground. Snowball A is
  thrown straight downward snowball B is thrown in
  a direction 25o above the horizontal. When the
  snowballs land, is the speed of A greater than,
  less than, or the same speed of B? Verify your
  answer by calculation of the landing speed of
  both snowballs.

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Projectiles launched over level ground

• These projectiles have highly symmetric
  characteristics of motion.
• It is handy to know these characteristics, since
  a knowledge of the symmetry can help in working
  problems and predicting the motion.
• Lets take a look at projectiles launched over
  level ground.

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Trajectory of a 2-D Projectile

• Definition The trajectory is the path traveled
  by any projectile. It is plotted on an x-y graph.

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Trajectory of a 2-D Projectile

• Mathematically, the path is defined by a parabola.

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Trajectory of a 2-D Projectile

• For a projectile launched over level ground, the
  symmetry is apparent.

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Range of a 2-D Projectile
Range

• Definition The RANGE of the projectile is how
  far it travels horizontally.

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Maximum height of a projectile
Maximum Height
Range

• The MAXIMUM HEIGHT of the projectile occurs when
  it stops moving upward.

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Maximum height of a projectile
Maximum Height
Range

• The vertical velocity component is zero at
  maximum height.

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Maximum height of a projectile
Maximum Height
Range

• For a projectile launched over level ground, the
  maximum height occurs halfway through the flight
  of the projectile.

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Acceleration of a projectile

• Acceleration points down at 9.8 m/s2 for the
  entire trajectory of all projectiles.

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Velocity of a projectile
v
v
v
vo
vf

• Velocity is tangent to the path for the entire
  trajectory.

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Velocity of a projectile
vx
vx
vy
vy
vx
vy
vx
vy
vx

• The velocity can be resolved into components all
  along its path.

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Velocity of a projectile
vx
vx
vy
vy
vx
vy
vx
vy
vx

• Notice how the vertical velocity changes while
  the horizontal velocity remains constant.

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Velocity of a projectile
vx
vx
vy
vy
vx
vy
vx
vy
vx

• Maximum speed is attained at the beginning, and
  again at the end, of the trajectory if the
  projectile is launched over level ground.

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Velocity of a projectile

• Launch angle is symmetric with landing angle for
  a projectile launched over level ground.

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Time of flight for a projectile

• The projectile spends half its time traveling
  upward

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Time of flight for a projectile

• and the other half traveling down.

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Problem 21, Homework

• A basketball is thrown horizontally with a speed
  of 4.0 m/s. A straight line drawn from the
  release point to the landing point makes an angle
  of 30.0o with the horizontal. What was the
  release height?

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Problem 23, Homework

• A second baseman tosses the ball (speed 17 m/s at
  35o above horizontal) to the first baseman, who
  catches it at the same level at which it was
  thrown.
• A) What is the horizontal component of the balls
  velocity just before it was caught?
• B) How long is the ball in the air?

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Problem 25, Homework

• A cork shoots out of a champagne bottle at an
  angle of 40.0o above the horizontal. If the cork
  travels a horizontal distance of 1.50 m in 1.25
  s, what was its initial speed?

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Problem 27, Homework

• A basketball forward makes a bounce pass to the
  center. The ball is thrown with an initial speed
  of 4.3 m/s at an angle of 15o below the
  horizontal. It is released 0.80 m above the
  floor. What horizontal distance does the ball
  cover before bouncing?

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Problem 28, Homework

• Repeat the previous problem for a bounce pass in
  which the ball is thrown 15o above the horizontal.

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Questions, Homework

• 2. A projectile is launched over level ground
  with speed vo and an angle of q above the ground.
  What is the average velocity between launch and
  landing?
• 3. A projectile is launched from level ground.
  When it lands, its direction of motion has
  rotated clockwise through 60o. What was the
  launch angle?

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Position graphs for 2-D projectiles
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Velocity graphs for 2-D projectiles
Vy
Vx
t
t
52
Acceleration graphs for 2-D projectiles
ay
ax
t
t
53
The Range Equation

• Derivation is an important part of physics.
• Your book has many more equations than your
  formula sheet.
• The Range Equation is in your textbook, but not
  on your formula sheet. You can use it if you can
  memorize it or derive it!

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The Range Equation

• R (vo2/g)sin2q.
• R range of projectile fired over level ground
• vo initial velocity
• g acceleration due to gravity
• q launch angle

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Sample problem

• A golfer tees off on level ground, giving the
  ball an initial speed of 42.0 m/s and an initial
  direction of 35o above the horizontal.
• How far from the golfer does the ball land?
• The next golfer hits a ball with the same initial
  speed, but at a greater angle than 45o. The ball
  travels the same horizontal distance. What was
  the initial direction of motion?

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Projectile Lab

• The purpose is to collect data to plot a
  trajectory for a projectile launched
  horizontally, and to calculate the launch
  velocity of the projectile. Equipment is
  provided, you figure out how to use it.
• What you turn in
• a table of data
• a graph of the trajectory
• a calculation of the launch velocity of the ball
  obtained from the data
• Hints and tips
• The thin paper strip is pressure sensitive.
  Striking the paper produces a mark.
• You might like to hang a sheet of your own graph
  paper on the brown board.

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Sample Problem

• Playing shortstop, you throw a ball horizontally
  to the second baseman with a speed of 22 m/s. The
  ball is caught by the second baseman 0.45 s
  later.
• How far were you from the second baseman?
• What is the distance of the vertical drop?

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Announcements 11/19/2009

• Homework collected tomorrow
• but none to turn in
• Exam Tomorrow turn in classwork packet
• 2-D Kinematics.
• Projectile motion
• Exam Review tomorrow 700 AM
• Clicker Quiz (get out your classwork packet so I
  can check the back page)
• Free Response Review
• Sample problems.

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Sample Problem

• A golfer tees off on level ground, giving the
  ball an initial speed of 42.0 m/s and an initial
  direction of 35o above the horizontal.
• How far from the golfer does the ball land?
• The next golfer hits a ball with the same initial
  speed, but at a greater angle than 45o. The ball
  travels the same horizontal distance. What was
  the initial direction of motion?