Uniform Circular Motion & Relative Velocity

Seatwork #2• A man trapped in a valley desperately fires a

signal flare into the air. The man is standing 300.0 m from the base of a vertical 250.0m cliff when he fires the flare with an initial velocity of 100 m/s at an angle 55.0o from the horizontal. (a) How long does the flare stay in the air? (2 pts) (b) What is the distance from the man and the landing point of the flare? (2 pts) (c) What is the maximum height of the flare? (1 pt) [Ignore air resistance and assume man is very very short. Also assume ground at the top of the cliff is level.]

Other cases for 2D motion at constant acceleration•Uniform Circular Motion is defined as a

particle moving at constant speed, in a circle.

•

Acceleration on a Curve

•

0t

0v

1t

1v

dtvd

tva

tva

t

ave

0lim

0v

1v

v

avea

dtvd

tva

tva

t

ave

0lim

Similar Triangles

dtvd

tva

tva

t

ave

0lim

rrvv

vv

rr

trrva

tva

t

t

0

0

lim

lim

tr

rva

t

0lim

Since v and r are constant

trrva

tva

t

t

0

0

lim

lim

tr

rva

t

0lim

rva2

Acceleration on a Curve

•

0t

0v

2t

2v

0a

Magnitude of a is constant. But direction is changing and is always pointing inward.

We call this kind of acceleration Centripetal Acceleration

rva2

Example•A sports car has a lateral acceleration of

0.96g’s. This is the maximum centripetal acceleration it can attain without skidding out of a circular path. If the ca is travelling at a constant 40m/s, what is the maximum radius of curve it can negotiate?

Example

mavr

rva

170408.9)40( 22

2

2408.9)8.9)(96.0(96.0 smga

Uniform Circular Motion•Will be discussed in more depth later on

in the semester.

Relative Velocity•So far we’ve discussed velocity relative to

a stationary point.•What happens then if an observer is

moving?

A simple example• A man is walking at a rate

of 1.2 m/s (in the +x direction) on a moving train that has a speed of 15.0 m/s (+x direction)

• What is the velocity of the man?

A simple example• A man is walking at a rate of

1.2 m/s on a moving train that has a speed of 15m/s

• What is the velocity of the man?

• 2 observers, someone on the train (A) and someone off the train (B).

• A will say the man is moving at 1.2 m/s

• B will say the man is moving at 16.2 m/s

• Two observers have different frames of reference

Relative Velocity in One Direction• 2 observers, someone on

the train (A) and someone off the train (B). Denote passenger as (P).

• We shall define

• = the velocity of C relative to the frame of D

DCv |

Relative Velocity in One Direction• 2 observers, someone on

the train (A) and someone off the train (B). Denote passenger as (P).

• We shall define

• = the velocity of C relative to the frame of D

DCv |

sm

APv 2.1|

sm

BAv 0.15|

Relative Velocity in One Direction• = the velocity of C

relative to the frame of DDCv |

sm

APv 2.1|

sm

BAv 0.15|

sm

BP

BP

BAAPBP

v

v

vvv

2.16

152.1

|

|

|||

Relative Velocity in One Direction

• = the velocity of D relative to the frame of C

• = the velocity of C relative to the frame of D

DCv |

CDv |

CDDC vv ||

Another Example• You are driving north on a

straight two lane road at a constant 88 kph. A truck is travelling at 104kph approaching you (on the other lane). (a) What is the trucks velocity relative to you? (b) What is your velocity relative to the truck? (c) How do the relative velocities change after you and the truck pass?

Another Example• Denote T for truck, Y for

you• We need a third observer

so let it be the Earth E.• Given

• Find

hkm

ET

hkm

EC

v

v

104

88

|

|

?

?

|

|

TC

CT

v

v

Another Example•

hkm

ET

hkm

EC

v

v

104

88

|

|

sm

TC

sm

CT

CT

ECETCT

CEETCT

v

v

v

vvv

vvv

192

192

88104

|

|

|

|||

|||

Another Example• Do the relative velocities

change when the truck passes you?

Lets complicate matters• A man is walking at a rate

of 1.2 m/s (in the +z) direction on a moving train that has a speed of 15.0 m/s (+x direction)

• What is the velocity of the man?

Lets complicate matters• A man is walking at a rate

of 1.2 m/s (in the +z) direction on a moving train that has a speed of 15.0 m/s (+x direction)

• What is the velocity of the man?

sm

sm

BP

BP

BAAPBP

v

v

vvv

1505.15

)15()2.1(

|

22|

|||

Lets complicate matters• Instead of walking, the

passenger on the train throws a ball straight up into the air and catches is. What path does it take for the passenger? For the observer off the train?

• Revisit the military helicopter that accidentally dropped a bomb. For the helicopter what was the motion of the bomb? For an observer on the ground?

Example• A boat heading due north

crosses a wide river with a speed of 10kph relative to the water. The water has a speed of 5.0kph due east relative to the earth. Determine the velocity of the boat relative to the earth. (Compute up to 2 sig figs)

Example

ijv

vvv

iv

jv

hkm

hkm

EB

ERRBEB

hkm

ER

hkm

RB

ˆ5ˆ10

ˆ5

ˆ10

|

|||

|

|

Example

kphv

kphv

jiv

EB

EB

hkm

hkm

EB

11

18.11105

ˆ10ˆ5

|

22|

|

Example

2756.26

105tantan

tan

1

|

|1

|

|

RB

ER

RB

ER

vv

vv

Young and Freedman Problem 3.40• A pilot wishes to fly due

west. A wind of 80.0 km/h is blowing due south. If the speed of the plane in still air is 320 km/h, (a) in which direction should the pilot head? (b) what is the speed of the plane over the ground?