Transcript Document 7866145

Relative Velocity
Airplane Velocity Vectors
v plane , ground  v plane ,air  vair , ground
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
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
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
Relative Motion...
Vp,g = Vp,a + Va,g
 is a vector equation relating the
airplane’s velocity in different reference
frames.
V
a,g
Vp,a
Vp,g
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?
Va,g
Vp,a
Vp,g
• 102 km/h, 79o
Boat in River Velocity
vboat ,shore  vboat ,water  vwater ,shore
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 .
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?
11%
26%
63%
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?
Perpendicular Velocities ACT
Vx = 2 m/s
1m
Vy = 0.5 m/s
• How long does it take the ladybug to crawl to
the opposite side of the paper?
1m
t

2
s
0.5 ms
This is independent of vx!!!!!!!!!!
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
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)
50
(b) 50
(c)
3  29
2  35
50 1  50
50 m
1 m/s
2 m/s
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 that
your 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
x
2 2 12
 3 m/s
1m/s
solution

So the y component of your velocity with respect to the water is

So the time to get across is
50 m
 29s
3m s
3 m/s
50 m
y
x
3 m/s
Frame of Reference