2D and 3D motion

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Lecture 5

• 2D and 3D motion

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ACT Alice and Bob
Alice and Bob stand at the top of a cliff of
height h. Both throw a ball with initial speed
v0, Alice straight up and Bob straight down. The
speed of the balls when they hit the ground are
vA and vB, respectively. Which of the following
is true?

A) vA lt vB B) vA vB C) vA gt vB
v0
v0
y y0 h v2 v02 2gh same for both!
vA
vB
Alice Bob
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Most importantly symmetry! When the Alices ball
passes it initial position, its velocity is v0
pointing down (just like Bobs)
v0
v0
v0
vA
vB
Alice Bob
vA vB
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2D (and 3D) motion

• Now we need vectors to indicate position,
  velocity and acceleration, but the definitions we
  use in 1D are pretty much the same.

Position


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Velocity
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Acceleration

• Average
•  
• Instantaneous
•  

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ACT Acceleration
Shown below are the trajectory of a moving
object and the snapshots taken every second.
Which of the following is true about the
components of the acceleration?
y
4s
3s
2s
x
1s
A) ax 0, ay gt 0 B) ax gt 0, ay gt 0 C)
ax lt 0, ay 0
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Note Both the speed and the direction of
velocity are changing!
y
4s
3s
2s
v(3)
x
1s
v(2)
v(1)
v(1)
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The big new thing in 2D changes in direction
An object can move at constant speed and still
have a ?0! This didnt happen in 1D!!


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In 2 (or 3) dimensions, acceleration can occur
both parallel to velocity or perpendicular to it
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Constant acceleration

• Same equations as in 1D but vectorial.

(3 equations each)
(This is now a dot-product)
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Projectiles
This an important example of a 2D problem.

• We will consider projectiles in free fall where
  we neglect air resistance and curvature of the
  Earth.
• With the most common choice of coordinate system
• ay ? g
• ax 0, vx constant!

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Example Projectiles
A projectile is fired from a cannon at a
30-degree angle with the ground and an initial
velocity of 100 m/s. Assuming no air resistance
and g 10 m/s2, calculate the time it will spend
in the air.
a. 2.5 s b. 5.0 s c. 10 s d. 20 s e. 40
s
yfinal 0
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ACT Shoot the monkey(tranquilizer gun)

• A zookeeper shoots a tranquilizer dart to a
  monkey that hangs from a tree. He aims at the
  monkey and shoots a dart with an initial speed
  v0. The monkey, startled by the gun, lets go
  immediately. Will the dart hit the monkey?
• Only if v0 is large enough.
• Yes, regardless of the magnitude of v0.
• No, it misses the monkey.

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If there is no gravity, the dart hits the monkey
rr0
rv0t
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If there is gravity, the dart also hits the
monkey!
rr0 gt2/2
rv0t - gt2/2
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This might be easier to think about
x v0t y - gt2/2
x x0 y - gt2/2
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EXAMPLE Mark McGwire.

• Mark McGwire clobbers a fastball toward
  center-field. The ball is hit 1 m above the
  plate, and its initial velocity is 36.5 m/s at an
  angle of 30 above horizontal. The center-field
  wall is 113 m from the plate and is 3 m high.
• a. How long after the hit does the ball reach the
  fence?

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Mark McGwire clobbers a fastball toward
center-field. The ball is hit 1 m above the
plate, and its initial velocity is 36.5 m/s at an
angle of 30 above horizontal. The center-field
wall is 113 m from the plate and is 3 m high.
a. How long after the hit does the ball reach the
fence?
Draw a figure and select axes.
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Mark McGwire clobbers a fastball toward
center-field. The ball is hit 1 m above the
plate, and its initial velocity is 36.5 m/s at an
angle of 30 above horizontal. The center-field
wall is 113 m from the plate and is 3 m high.
We need an equation that relates final position
to time.
but ax0
Checks Units Reasonable Limit ?
90
ok
ok
ok
It takes 3.58s to reach the fence assuming that
it has not collided with the ground. To fully
resolve this issue we move to part B.
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Mark McGwire clobbers a fastball toward
center-field. The ball is hit 1 m above the
plate, and its initial velocity is 36.5 m/s at an
angle of 30 above horizontal. The center-field
wall is 113 m from the plate and is 3 m high.
b. Does Mark get a home run?
When x 113 m, is y gt 3m?
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Mark McGwire clobbers a fastball toward
center-field. The ball is hit 1 m above the
plate, and its initial velocity is 36.5 m/s at an
angle of 30 above horizontal. The center-field
wall is 113 m from the plate and is 3 m high.
Lets find the value of y at t 3.58 s We need a
relation which describes the evolution of y as a
function of time
ok
Checks Units
Reasonable
ok
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Mark McGwire clobbers a fastball toward
center-field. The ball is hit 1 m above the
plate, and its initial velocity is 36.5 m/s at an
angle of 30 above horizontal. The center-field
wall is 113 m from the plate and is 3 m high.
c. What is the speed of the ball as it hits the
ground?
Hit the ground y 0
To find v, we need a relation between y and the
final velocity
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Mark McGwire clobbers a fastball toward
center-field. The ball is hit 1 m above the
plate, and its initial velocity is 36.5 m/s at an
angle of 30 above horizontal. The center-field
wall is 113 m from the plate and is 3 m high.
An alternative (and waaaaay longer) approach
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THE END
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Of course we have neglected air resistance. This
would lower the trajectory of the ball