Kinematics in Two Dimensions

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Kinematics in Two Dimensions

• Position, velocity, acceleration vectors
• Constant acceleration in 2-D
• Free fall in 2-D
• Serway and Jewett 4.1 to 4.3

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y
The Position vector points from the origin
to the particle.
path
(x,y)
yj
xi
x
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(No Transcript)
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Acceleration is the rate of change of velocity
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• a is the rate of change of v (Recall a
  derivative gives the rate of change of function
  wrt a variable, like time).
• Velocity changes if
• i) speed changes
• ii) direction changes (even at constant speed)
• iii) both speed and direction change
• In general, acceleration is not parallel to the
  velocity.

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Concept Quiz
A pendulum is released at (1) and swings across
to (5).
a) at 3 only b) at 1 and 5 only c) at 1, 3, and
5 d) none of the above
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Components Each vector relation implies 3
separate relations for the 3 Cartesian components.
(i, j, k, are unit vectors)
We get velocity components by differentiation
the unit vectors are constants
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Each component of the velocity vector looks like
the 1-D velocity we saw earlier. Similarly for
acceleration
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Common Notation for time derivatives only, a
dot is often
used
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Constant Acceleration Projectile Motion
In 2-D, each vector equation is equivalent to a
pair of component equations
Example
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Shooting the Gorilla
Tarzan has a new AK-47. George the gorilla hangs
from a tree branch, and bets that Tarzan cant
hit him. Tarzan aims at George, and as soon as he
shoots his gun George lets go of the branch and
begins to fall. Where should Tarzan be aiming
his gun as he fires it?A) above the gorillaB)
at the gorillaC) below the gorilla
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ag
v0t
(1/2)gt2
v0
r0
r(t) r0v0t (1/2)gt2
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Concept quiz

• Your summer job at an historical site includes
  firing a cannon to amuse tourists.
  Unfortunately, the cannon isnt properly
  attached, and as the cannonball shoots forward
  (horizontally) the cannon slides backwards off
  the wall.
• If the cannon hits the ground 2 seconds later,
  the cannonball will hit the ground

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Example Problem
A stone is thrown upwards from the top of a 45.0
m high building with a 30º angle above the
horizontal. If the initial velocity of the stone
is 20.0 m/s, how long is the stone in the air,
and how far from the base of the building does it
land ?
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Example Problem Cannon on a slope.
How long is the cannonball in the air, and how
far from the cannon does it hit? Try to do it two
different ways once using horizontal and
vertical axes, once using axes tilted at 20o.
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Show that for
d
vo
?
F
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Summary

• position vector points from origin to a
  particle
• velocity vector
• acceleration vector
• for constant acceleration, we can apply 1-D
  formulae to each component separately
• for free fall in uniform , horizontal and
  vertical motions are independent