Two Dimensional Dynamics

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Two Dimensional Dynamics
CPI Lecture 06
Exam I

• Todays lecture will cover Textbook Sections
  3.1-3.5

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Brief Review Thus Far

• Newtons Laws of motion
• Kinematics
• Dynamics
• Today we work in 2 Dimensions!

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2-Dimensions

• X and Y are INDEPENDENT!
• Break 2-D problem into two 1-D problems

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Equilibrium
Determine the tension in the 6 meter rope if it
sags 0.12 meters in the center when a gymnast
with weight 250 Newtons is standing on it.
x direction -TL cosq TR cosq 0 TL TR
y direction TL sinq TR sinq - W 0 2 T sinq
W T W/(2 sinq) 3115 N
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Position, Velocity and Acceleration

• Position, Velocity and Acceleration are Vectors!
• x and y directions are INDEPENDENT!

y direction
x direction
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Velocity in Two Dimensions
A ball is rolling on a horizontal surface at 5
m/s. It then rolls up a ramp at a 25 degree
angle. After 0.5 seconds, the ball has slowed to
3 m/s. What is the change in velocity?
x-direction vix 5 m/s vfx 3 m/s cos(25) Dvx
3cos(25)5 -2.28m/s
y-direction viy 0 m/s vfy 3 m/s sin(25) Dvy
3sin(25)1.27 m/s
3 m/s
5 m/s
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Acceleration in Two Dimensions
A ball is rolling on a horizontal surface at 5
m/s. It then rolls up a ramp at a 25 degree
angle. After 0.5 seconds, the ball has slowed to
3 m/s. What is the average acceleration?
x-direction
y-direction
3 m/s
5 m/s
25
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Kinematics in Two Dimensions

• x x0 v0xt 1/2 axt2
• vx v0x axt
• vx2 v0x2 2ax ?x

• y y0 v0yt 1/2 ayt2
• vy v0y ayt
• vy2 v0y2 2ay ?y

x and y motions are independent! They share a
common time t
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Kinematics for Projectile Motionax 0
ay -g

• y y0 v0yt - 1/2 gt2
• vy v0y - gt
• vy2 v0y2 - 2g ?y

• x x0 v0t
• vx v0x

X
Y
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Pre-Flight 34
A flatbed railroad car is moving along a track at
constant velocity. A passenger at the center of
the car throws a ball straight up. Neglecting
air resistance, where will the ball land? 1.
Forward of the center of the car 2. At the center
of the car 3. Backward of the center of the car
Demo - train
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Great Answers!
Since no air resistance is present, the ball and
the train would be moving with the same
horizontal velocity, and when the ball is tossed,
it is given an additional velocity component in
the vertical direction, but the original
horizontal velocity component remains unchanged,
and lands in the center of the train. Alas, that
is all!
I've done it before and it doesn't hit me in the
head.
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ACT
A flatbed railroad car is accelerating down a
track due to gravity. The ball is shot
perpendicular to the track. Where will it
land? 1. Forward of the center of the car 2. At
the center of the car 3. Backward of the center
of the car
X direction Ball mg sin(q) ma a g sin(q)
X direction Cart mg sin(q) ma a g sin(q)
Same acceleration gives same position
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Lecture 4, Pre-Flight 12

• You and a friend are standing on level ground,
  each holding identical baseballs. At exactly the
  same time, and from the same height, you drop
  your baseball without throwing it while your
  friend throws her baseball horizontally as hard
  as she can. Which ball hits the ground first?
• 1. Your ball
• 2. Your friends ball
• 3. They both hit the ground at the same time

Demo - marbles
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Pre-Flight 12 (great answers)
The y-variables for both baseballs are the same.
Both balls have an initial y velocity of 0, and
both accelerate at the same rate due to gravity.
Therefore, they hit the ground at the same time.
The balls will hit the ground at the same time
because the x and y components of motion are
independent of each other. My friend's ball has
positive x velocity while my ball has a no x
velocity, but the force acting on the ball in the
y direction (gravity) is equal in both instances.

• y y0 voyt - gt2/2
• v0y 0 and y0
• therefore tsqrt(2y0/g)
• result is independent of v0x

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Pre-Flight 56

• You are a vet trying to shoot a tranquilizer dart
  into a monkey hanging from a branch in a distant
  tree. You know that the monkey is very nervous,
  and will let go of the branch and start to fall
  as soon as your gun goes off. On the other hand,
  you also know that the dart will not travel in a
  straight line, but rather in a parabolic path
  like any other projectile. In order to hit the
  monkey with the dart, where should you point the
  gun before shooting?
• 1 Right at the monkey
• 2 Below the monkey
• 3 Above the monkey

Demo - monkey
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See text 4-3
Shooting the Monkey...
x v0 t y -1/2 g t2
x x0 y -1/2 g t2
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Shooting the Monkey...
y y0 - 1/2 g t2

• Still works even if you shoot upwards!

y v0y t - 1/2 g t2
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Summary of Concepts

• X and Y directions are Independent!
• Position, velocity and acceleration are vectors
• F m a applies in both x and y direction
• Projective Motion
• ax 0 in horizontal direction
• ay g in vertical direction

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