Download - Uniform Circular Motion & Relative Velocity
![Page 1: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/1.jpg)
Uniform Circular Motion & Relative Velocity
![Page 2: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/2.jpg)
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.]
![Page 3: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/3.jpg)
Other cases for 2D motion at constant acceleration•Uniform Circular Motion is defined as a
particle moving at constant speed, in a circle.
•
![Page 4: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/4.jpg)
Acceleration on a Curve
•
0t
0v
1t
1v
dtvd
tva
tva
t
ave
0lim
0v
1v
v
avea
![Page 5: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/5.jpg)
dtvd
tva
tva
t
ave
0lim
![Page 6: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/6.jpg)
Similar Triangles
dtvd
tva
tva
t
ave
0lim
rrvv
vv
rr
![Page 7: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/7.jpg)
trrva
tva
t
t
0
0
lim
lim
tr
rva
t
0lim
Since v and r are constant
![Page 8: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/8.jpg)
trrva
tva
t
t
0
0
lim
lim
tr
rva
t
0lim
rva2
![Page 9: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/9.jpg)
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
![Page 10: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/10.jpg)
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?
![Page 11: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/11.jpg)
Example
mavr
rva
170408.9)40( 22
2
2408.9)8.9)(96.0(96.0 smga
![Page 12: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/12.jpg)
Uniform Circular Motion•Will be discussed in more depth later on
in the semester.
![Page 13: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/13.jpg)
Relative Velocity•So far we’ve discussed velocity relative to
a stationary point.•What happens then if an observer is
moving?
![Page 14: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/14.jpg)
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?
![Page 15: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/15.jpg)
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
![Page 16: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/16.jpg)
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 |
![Page 17: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/17.jpg)
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|
![Page 18: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/18.jpg)
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
|
|
|||
![Page 19: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/19.jpg)
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 ||
![Page 20: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/20.jpg)
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?
![Page 21: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/21.jpg)
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
![Page 22: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/22.jpg)
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
|
|
|
|||
|||
![Page 23: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/23.jpg)
Another Example• Do the relative velocities
change when the truck passes you?
![Page 24: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/24.jpg)
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?
![Page 25: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/25.jpg)
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|
|||
![Page 26: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/26.jpg)
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?
![Page 27: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/27.jpg)
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)
![Page 28: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/28.jpg)
Example
ijv
vvv
iv
jv
hkm
hkm
EB
ERRBEB
hkm
ER
hkm
RB
ˆ5ˆ10
ˆ5
ˆ10
|
|||
|
|
![Page 29: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/29.jpg)
Example
kphv
kphv
jiv
EB
EB
hkm
hkm
EB
11
18.11105
ˆ10ˆ5
|
22|
|
![Page 30: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/30.jpg)
Example
2756.26
105tantan
tan
1
|
|1
|
|
RB
ER
RB
ER
vv
vv
![Page 31: Uniform Circular Motion & Relative Velocity](https://reader033.vdocument.in/reader033/viewer/2022061615/56815efd550346895dcdbadf/html5/thumbnails/31.jpg)
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?