Relative Velocity

Airplane Velocity Vectors

, , ,plane ground plane air air groundv v v

Relative Motion...• The plane is moving north in the frame

of reference attached to the air:– VVp, a is the velocity of the plane w.r.t. the

air.AirAir

VVp,a

Relative Motion...• But suppose the air is moving east in

the IRF attached to the ground.– VVa,g is the velocity of the air w.r.t. the

ground (i.e. wind).

VVa,g

AirAir

VVp,a

Relative Motion...• What is the velocity of the plane in a

frame of reference attached to the ground?– VVp,g is the velocity of the plane w.r.t. the

ground.

VVp,g

Relative Motion...VVp,g = VVp,a + VVa,g is a vector equation relating the

airplane’s velocity in different reference frames.

VVp,g

VVa,g

VVp,a

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?

VVp,g

VVa,g

VVp,a

• 102 km/h, 79o

Boat in River Velocity

, , ,boat shore boat water water shorev v v

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?

26%

11%

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

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

1 m

Vx = 2 m/s

Vy = 0.5 m/s

12

0.5 ms

mt s 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)

(b)

(c)

50 3 29

2 m/s

1 m/s50 m

50 2 35

50 1 50

solution solution

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

Choose x axis along riverbank and y axis across rivery

x

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:

2 m/s 1m/sy

x

1 m/s

2 1

3

2 2

m/s

solutionsolution

So the y component of your velocity with respect to the water is

So the time to get across is

y

x

3m/s

503

29m

m ss

50 m

3m/s

Frame of Reference