Physics 2211 Mechanics Lecture 10 Knight: 6'1 to 6'4 TwoDimensional Motion

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Physics 2211 - MechanicsLecture 10 (Knight
6.1 to 6.4)Two-Dimensional Motion

• Dr. John Evans

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Kinematics in Two Dimensions
Roller Coaster Ride
y
x
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Position and Velocity
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Position-Time vs x-y Graphs
s-t Graph Slope velocity
x-y Graph Slope local trajectory
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Graphs of x-y and v-t
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Acceleration in x-y Graphs
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Constant Acceleration
The x and y motions are independent, but both
depend on t.
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Example Shuttle Takeoff
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Dynamics in Two Dimensions
Newtons 2nd Law

• Draw a pictorial representation and a physical
  representation (motion diagram and free-body
  diagram).
• Use Newtons second law in component form

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Example Rocketing Puck (1)
Alice tapes a 0.2 kg model rocket that
generates 8.0 N of thrust to a 0.4 kg ice hockey
puck. She orients the puck so that the rockets
nose points in the y direction, then pushes the
puck across the ice in the x direction. She
releases it with a speed of 2.0 m/s at the exact
instant when the rocket fires. Find an
equation for the systems trajectory and draw a
graph of it.
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Example Rocketing Puck (2)
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Projectile Motion
A projectile that is launched with an initial
velocity (vix,viy) follows a parbolic
trajectory. vixvicosq and viyvisinq
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Reasoning about Projectile Motion
A hungry hunter wants to shoot down a
coconut that is hanging from the branch of a palm
tree. He aims the gun directly at the coconut,
but, as luck would have it, the coconut falls
from the branch at the exact instant that the
hunter pulls the trigger. Does the bullet
hit the coconut? YES! Because the bullet
and coconut both fall a distance ½gt2 in time t.
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Projectile Motion Solutions
PROBLEM-SOLVING STRATEGY 6.1 Projectile motion
problems MODEL Make simplifying
assumptions. VISUALIZE Use a pictorial
representation. Establish a coordinate system
with the x-axis horizontal and the y-axis
vertical. Show important points in the motion on
a sketch. Define symbols and identify what the
problem is trying to find. SOLVE The acceleration
is known and
. Thus the problem becomes one of kinematics.
The kinematic equations are

is the same for the horizontal and
vertical components of the motion. Find
from one component, then use that value for the
other component. ASSESS Check that your result
has the correct units, is reasonable, and answers
the question.
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Example A Home Run
A baseball is hit so that it leaves the bat
making a 300 angle with the ground. It crosses a
low fence at the boundary of the ballpark 100 m
from home plate at the same height that it was
struck. (Neglect air resistance.) What was
its velocity as it left the bat?
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Projectile Range
Distance v02sin(2q)/g.
Note that sin(2q)1 at q450, so distance is
maximum there. Conclusion in the absence
of air resistance, a projectile launched at 450
will travel farthest before returning to the the
height of the launch.
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Relative Motion
Amy, Bill, and Carlos all measure the
velocity of the runner and the acceleration of
the jet plane. The green velocity vectors are
shown in Amys reference frame. Amy vR5
m/s Bill vR0 m/s What about
aplane? Carlos vR-10 m/s
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Relative Position
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Example A Ball Toss

• Mike throws a ball upward at a 630 angle
  with a speed of 22 m/s. Nancy rides past Mike on
  her bicycle at 10 m/s at the instant he releases
  the ball.
• What trajectory does Mike see?
• What trajectory does Nancy see?

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Object and Frame VelocitiesAdd Vectorially
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Galilean Relativity (1)
Question In which reference frames are Newtons
Laws valid? Does in all reference
frames? Suppose that a net force Fnet acts
on a object in reference frame S, and that
Newtons Laws have been tested and found valid in
S. Then experimenters in S will find that the
object is observed to have acceleration a such
that . Now consider reference
frame S, which moves relative to frame S with
velocity v. In S, does ? To
answer this question, we must transform F and a
to the new frame. The force strength F does
not depend on coordinate system. If a force
produces a spring scale reading in one frame, all
observers will see the same scale reading.
Therefore, F F.
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Galilean Relativity (2)
Now consider how the acceleration transforms
from S to S, which is moving with constant
velocity V relative to S. Velocities add, so
Galilean Relativity Newtons laws of motion are
valid in all inertial reference frames.
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Galileo vs. Einstein
The laser beam moves along the x axis away
from Tom at the speed of light, vx3 x 108 m/s.
Sue flies by in her space ship, moving along
the x axis at Vx2 x 108 m/s. From her point of
view, how fast is the laser beam
moving? Galileo vx vx Vx 1 x 108
m/s Einstein vx vx 3 x 108 m/s
Newtons laws of motion are valid in all inertial
reference frames.
The speed of light is the same in all inertial
reference frames.
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End of Lecture 10

• For next time, read Chapters 7.1 through 7.5.