Terminal Velocity

 

Remember that there are two primary forces that act on falling objects in the laboratory.

  1. gravity: (F force, m mass, g acceleration due to gravity)
  2. drag force: (force due to the air on the falling mass)

 

One can derive an expression for F2 based on a model for the wind resistance. Experience tells us that drag will change when the object changes its speed. Holding your hand out the window of a moving car shows us that the faster the car travels the more force you feel from the air pushing on your hand (greater air drag). Also the area exposed will change the drag force. The gravitational force should depend only on the mass. One can also argue that the wind or drag force will not depend on the mass.

 

One common model for the drag forces for falling objects is:

 

(the negative sign directs the force opposite to the velocity )

 

The model specifically requires that the force is proportional to the velocity squared.

Setting the forces equal (and opposite) as the condition for when terminal velocity is reached, we obtain

 

 

One expression [eq.1] is written so that the mass (dependent) depends on a function of the terminal velocity (independent) and another rewritten so that the terminal velocity (dependent) depends on the mass (independent). You should see that both equations are the equivalent just rearranged using standard algebra.

 

The only unknown is the drag coefficient. So the data must be matched to the model by adjusting C.

 

A different approach might be not to assume as specific a formula. Let us consider more options and let the force depend on some power of the velocity. The function then has two unknown parameters and as follows:

 

 

Equating forces (equal but opposite) as the condition when the filters fall at constant speed we obtain

 

 

With eq. 3 and eq. 4 the relevant equations that model the process.

 

 

 

In analyzing filter data

 

In this lab we will use equation 1 and 3 and examine how the mass depends on the terminal velocity.