Physics 1710 Chapter 4 Motion in Two Dimensions

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Physics 1710Chapter 4 Motion in Two Dimensions

• Quiz
• Two identical row boats that can be rowed at 3.0
  m/s are both attempting to navigate a small
  river. If one row boat goes upstream against the
  current (2.0 m/s) for 100 m and then returns
  while a second rows across the stream for 100 m
  and returns to the starting point, which will be
  the the winner in this race?

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Physics 1710Chapter 4 Motion in Two Dimensions
t1 d/ (vo V ) d/ (vo V)
2
t1 100/ (3-2) 100/ (32) 120 sec
V
t2 2d/ (vvo2 V 2 )
t2 2(100)/ (v32 22 ) 200/ (v5 )
89.4 sec
v vvo2 V 2
1
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Physics 1710Chapter 4 Motion in Two Dimensions

• 1 Lesson
• Displacement, velocity and acceleration are
  vector quantities in two or more dimensions.
• Each component of the kinematic variables
• is separate and independent.
• In projectile motion the x-motion is
  unaccelerated while the y-motion experiences a
  constant acceleration equal to g.
• Uniform circular motion requires an
  ever-changing acceleration of constant magnitude.
• Relative motion obeys the Galilean
  Transformation.

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Physics 1710Chapter 4 Motion in Two Dimensions

• Position r is a vector.
• r x i y j
• Or rx x ry y
• Thus, one can represent a vector by a position
  vector, ie an arrow.

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Physics 1710Chapter 4 Motion in Two Dimensions

• Displacement in 2-D is a vector
• ? r r final - r initial
• ? rx x final - x initial ?x
• ? ry yfinal - y initial ?y
• ? r ?? r?
• ?? r? v(x final - x initial ) 2 (yfinal - y
  initial) 2
• Tan ? (yfinal - y initial)/ (x final - x
  initial )

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Physics 1710Chapter 4 Motion in Two Dimensions

• Average velocity is a vector
• vave ? r / ?t
• This means the following
• vx, ave ? x / ?t
• vy, ave ? y / ?t .

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Physics 1710Chapter 4 Motion in Two Dimensions

• Instantaneous velocity is a vector
• v lim ?t ? 0 ? r / ?t
• This means the following
• vx lim ?t ? 0 ? x / ?t dx /dt
• vy lim ?t ? 0 ? y / ?t dy/dt

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Physics 1710Chapter 4 Motion in Two Dimensions

• Average acceleration is a vector
• aave ? v / ?t
• This means the following
• ax, ave ?vx / ?t
• ay, ave ?vy / ?t .

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Physics 1710Chapter 4 Motion in Two Dimensions

• Instantaneous acceleration is a vector
• a lim ?t ? 0 ? v / ?t
• This means the following
• ax lim ?t ? 0 ? vx / ?t dvx /dt
• ay lim ?t ? 0 ? vy / ?t dvy /dt
• N.B.
• The components are strictly segregated!

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Physics 1710Chapter 4 Motion in Two Dimensions

• Motion in Two Dimensions
• All the one dimensional kinematic equations can
  be generalized to two (or more) dimensions.
• All the component equations obey the one
  dimensional kinematics separately.

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Physics 1710Chapter 4 Motion in Two Dimensions

• Vector kinematic equations (uniform a)
• rfinal rinitial vinitial t ½ a t 2
• vfinal vinitial a t

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Physics 1710Chapter 4 Motion in Two Dimensions

• Projectile Motion
• Horizontal acceleration
• ax 0
• ? vx, final vx, initial
• xfinal x initial vx, initial t .
• Vertical acceleration
• ay -g
• ? vy, final vy, initial - g t
• yfinal y initial vy, initial t - ½ g t 2

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Physics 1710Chapter 4 Motion in Two Dimensions

• Parabolic Motion

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Physics 1710Chapter 4 Motion in Two Dimensions

• Uniform Circular Motion
• ?a? a,
• a constant value always pointing toward the
  center of the circle. Centripedal acceleration.
• a ax i a y j,
• where
• ax a cos ?
• ay a sin ?

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Physics 1710Chapter 4 Motion in Two Dimensions

• Uniform Circular Motion
• a v 2 /R,
• The Centripedal acceleration, where v is the
  tangential speed and R is the radius of the
  circle.
• v 2pR / T ,
• Where T is the period or time to make one
  revolution.
• a 4p 2R/ T 2

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Physics 1710Chapter 4 Motion in Two Dimensions

• Uniform Circular Motion

Rd ?/ dt vd ? dv dt/v Rd(dv/v)/ dt 2
vd2v/dt 2 v 2 /R
d ?
R
v
d ?
dv dt
v
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Physics 1710Chapter 4 Motion in Two Dimensions

• Uniform Circular Motion

g
a
a
g
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Physics 1710Chapter 4 Motion in Two Dimensions

• Relative Motion
• and the Galilean Transformation
• d r '/dt d r/dt - vo
• v ' v vo
• d v '/dt d v/dt d vo/dt
• a ' a ao

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Physics 1710Chapter 4 Motion in Two Dimensions

• Summary
• Kinematics in two (or more) dimensions obeys the
  same 1- D equations in each component
  independently.
• Projectiles follow a parabola.
• Uniform circular motion has an acceleration
  toward the center of a circle that is constant in
  magnitude a v 2/R.
• Frames of reference moving relative to each
  other obey the Galilean transformation.