Relative Velocity

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Relative Velocity
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Airplane Velocity Vectors
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Relative Motion...

• The plane is moving north in the frame of
  reference attached to the air
• Vp, a is the velocity of the plane w.r.t. the
  air.

Air
Vp,a
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Relative Motion...

• But suppose the air is moving east in the IRF
  attached to the ground.
• Va,g is the velocity of the air w.r.t. the
  ground (i.e. wind).

Air
Vp,a
Va,g
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Relative Motion...

• What is the velocity of the plane in a frame of
  reference attached to the ground?
• Vp,g is the velocity of the plane w.r.t. the
  ground.

Vp,g
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Relative Motion...

• Vp,g Vp,a Va,g
• is a vector equation relating the airplanes
  velocity in different reference frames.

Va,g
Vp,a
Vp,g
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Airplane ACT

• The velocity of an airplane relative to the air
  is 100 km/h, due north. A crosswind blows from
  the west at 20 km/h. What is the velocity of the
  plane relative to the ground?

• 102 km/h, 79o

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Boat in River Velocity
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Motorboat ACT

• Consider a motorboat that normally travels 10
  km/h in still water. If the boat heads directly
  across the river, which also flows at a rate of
  10 km/h, what will be its velocity relative to
  the shore?

• When the boat heads cross-stream (at right angles
  to the river flow) its velocity is 14.1 km/h, 45
  degrees downstream .

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Preflight Responses

• Three swimmers can swim equally fast relative to
  the water. They have a race to see who can swim
  across a river in the least time. Relative to the
  water, Beth (B) swims perpendicular to the flow
  of the river (shown by the horizontal arrow in
  the figure), Ann (A) swims upstream, and Carly
  (C) swims downstream. Which swimmer wins the race?

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26
63
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Boat Velocity

• (1) Which boat takes the shortest path to the
  opposite shore?
• (2) Which boat reaches the opposite shore first?
• (3) Which boat provides the fastest ride?

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Perpendicular Velocities ACT
Vx 2 m/s
1 m
Vy 0.5 m/s

• How long does it take the ladybug to crawl to the
  opposite side of the paper?

• This is independent of vx!!!!!!!!!!

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Independence of Velocities

• If a boat heads perpendicular to the current at
  20 m/s relative to the river, how long will it
  take the boat to reach the opposite shore 100 m
  away in each of the following cases?
• Current speed 1 m/s
• Current speed 5 m/s
• Current speed 10 m/s
• Current speed 20 m/s

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Swimmer ACT

• You are swimming across a 50m wide river in which
  the current moves at 1 m/s with respect to the
  shore. Your swimming speed is 2 m/s with respect
  to the water. You swim across in such a way that
  your path is a straight perpendicular line across
  the river.
• How many seconds does it take you to get across
  ?(a) (b)(c)

50 m
2 m/s
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solution
y
Choose x axis along riverbank and y axis across
river
x

• The time taken to swim straight across is
  (distance across) / (vy )

• Since you swim straight across, you must be
  tilted in the water so thatyour x component of
  velocity with respect to the water exactly
  cancels the velocity of the water in the x
  direction

1 m/s
y
2 m/s
m/s
x
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solution

• So the y component of your velocity with respect
  to the water is
• So the time to get across is

m/s
m/s
50 m
y
x
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Frame of Reference