Resource Lesson
Air Resistance: Terminal Velocity
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Background Information:
The amount of air resistance an object encounters is directly proportional to its surface area and velocity.
Terminal velocity
,
v
_{t}
, is achieved when the air resistance equals the object's weight and the object can no longer accelerate. It reaches a state of
dynamic equilibrium
.
Theory.
The air resistance any group of filters encounters is directly proportional to their cross-sectional area and to their instantaneous velocity. The formula that represents this relationship is:
Equation #1
where
k
is a constant that is proportional to the filters' cross-sectional area. At terminal velocity,
v
_{t}
, the filters are in dynamic equilibrium,
Equation #2
Solving equation #2 for v
_{t}
produces the result,
Equation #3
Since each group of filters has relatively small mass, this terminal velocity is reached almost immediately after the filters are released. Thus, for the most part, they are traveling the entire time at
v
_{t}
. This allows us to use the equation
Equation #4
to determine the time required for each group of filters to reach the ground.
After solving equation #4 for time,
both times can be set equal to each other, since both groups in each trial hit the ground simultaneously.
Equation #5
Rearranging equation #5,
Equation #6
Substituting the expression for
v
_{t}
in equation #3 into equation #6 produces,
Equation #7
In equation #7,
g
and
k
cancel,
Equation #8
Squaring both sides of equation #8 results in the relationship,
Equation #9
This result means that the distance required to achieve terminal velocity for two objects, both released from rest, having the same surface area is directly proportional to the square root of their masses.
If one object has 9 times the mass of a second object how will their distances required to reach terminal velocity compare?
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