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.