Download NOTES - Air Resistance

11/7/12
Gravity
Air Resistance
Free Fall
•  An object is in free fall when the
only force on it is the force of
gravity.
•  Objects in free fall do not
experience air resistance.
•  Theoretically, objects in free fall
will continue to accelerate as
long as they are falling.
•  Gravity is NOT the force that pulls you down.
•  Gravity is an attractive force that attracts all
objects with mass inward towards other
objects with mass.
•  The force of gravity is not the same as the
acceleration due to gravity
Free Fall
•  All free falling objects
accelerate at the same
rate, regardless of their
mass (9.8 m/s2, or
approximately 10m/s2)
•  Newton’s 2nd Law
predicts this
a= F
m
Drag depends on:
Air Resistance or Drag
•  Air resistance is similar to friction in that
it opposes the direction of motion.
•  Air resistance is the resistance of the air
that the object is falling through.
•  Air resistance depends on two
variables:
–  The cross-sectional area of the object
–  The speed of the object.
•  Cross-Sectional Area
–  A small cross-sectional area
doesn’t encounter as much air
as a large cross-sectional area
does.
•  Speed:
–  When the object is moving
slowly, it is not moving through
much air so the air resistance
is small.
–  The faster the object moves,
the more air it travels through
and the air resistance is larger.
1
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Common Cross-Sectional Areas
Modeling Drag
D≈
€
•  D = Drag/air resistance, always
opposite the motion (N)
•  A = cross-sectional area (m2)
•  v = velocity of the object (m/s)
1 2
Av
4
Assumptions:
•  The object’s size (diameter) is between a few
millimeters and a few meters.
•  The objects speed is less than a few hundred meters per
second.
•  The object is moving through the air near Earth’s
surface (drag also depends on air density)
Book vs paper
•  If dropped, which will hit the ground first?
•  How does the drag force compare between the two?
–  It is the same (same size & velocity when dropped)
•  Why does the book hit first?
–  Larger force of gravity than paper
D
∑F
−mg + D
=
m
m
$
D'
a = −& g − )
% m(
a=
D
mg
mg
A cylinder falling
end down
A falling sphere
A box falling end
down
b
a
r
r
A = πr2
A = πr2
A = ab
L
A = 2rL
2r
A cylinder falling
side down
Which encounters the greater force of air resistance?
1. A falling elephant
2. A falling feather
Math: The elephant has a much larger
cross-sectional area (approximated as
a circle) than a feather. Therefore it will
experience a larger force of drag.
Concept: The elephant will “run into”
more air than the feather so he will
experience a larger drag force.
Result: The elephant has such a large
mass, the large drag force will not
effect it’s acceleration as much.
€
Draw the FBD of a baseball being thrown upward and
then caught with air resistance. (draw vectors to scale)
When the ball stops at the
top, there is no drag.
As the ball slows,
the drag decreases
mg
mg D
mg
Air Resistance
D
D
mg
D
As the ball
speeds up,
the drag
increases
Weight (mg)
mg
2
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When the forces are
balanced, acceleration
stops.
Air Resistance
Terminal Velocity
Air Resistance
Terminal Velocity:
• Air Resistance and weight
are balanced.
• Acceleration stops.
• Objects falls at a constant
velocity.
Weight (mg)
weight
Derive an equation for the terminal velocity of a
skydiver of mass m and cross-sectional area A
falling near the surface of the Earth.
D≈
∑ F = ma
D − mg = 0
1 2
Av ≈ mg
4
1 2
Av
4
€
€
€
v2 ≈
€
mg
v≈
4mg
A
4mg
A
€
€
A typical mass of a skydiver with their equipment is
75 kg. The average skydiver is 1.8m long and 0.40
m wide with a cross-sectional area of 0.72m2. What
is the typical terminal velocity when skydiving?
4mg
A
4(75kg)(9.8m / s 2 )
v≈
0.72m 2
v≈
€
v ≈ 64m / s
Or 143 mi/hr
The terminal speed of a Styrofoam ball is 15 m/s.
Suppose a Styrofoam ball is shot straight down with an
initial speed of 30 m/s. Which velocity graph is correct?
V (m/s)
1
V (m/s) 2
t
-15
-30
-30
3
t
-15
-15
-30
-30
V (m/s)
-15
V (m/s)
t
4
t
V (m/s)
5
t
-15
-30
€
€
3