Unicellular organisms such as bacteria and protists are small objects that often live in dense fluids (water, not air). Making sense of the movement of small things through water seems different from the movement of everyday objects in air (e.g. balls, rocks, cars on ramps). 

Consider the following conversation between two biology students:

Liz: I work in a microbiology lab, and in our experiments we are often manipulating the motion of objects suspended in a fluid. In such a fluid, to get objects to move at a constant speed, we have to apply a force that is proportional to the velocity of the object.

Jack: That can't be right. In physics, I learned that force is equal to the mass of the object times the acceleration of the object.

a.  In your group, first make sure you understand each perspective. What is Liz saying about the relationship between speed and force? What is Jack saying?

In the remainder of this problem we'll use the example of a paramecium swimming through water to further understand whether the perspectives of these two students are consistent or contradictory.

Paramecia move by pushing their cilia (little hairs on their surface) through the fluid. The fluid (of course) pushes back on them by Newton's third law. We will call this force of the fluid on the cilia of the paramecium "the applied force", Fapp (since it wouldn't happen if the paramecium didn't try to move its cilia). This is the force that moves the paramecium forward. The paramecium also feels a viscous force, which takes the form Fviscous = - βRv, proportional to the velocity and the effective radius of the object and in the opposite direction. (The "effective radius" of an object depends on its shape and its size somewhat, but for an object that is not too stretched out, we can take it to be some average radius of the object.)

b.  Draw a diagram that shows the forces acting on the paramecium as it moves.

c.  If the paramecium is moving at a constant velocity v_T, what is the applied force? Write an equation that could give a value for  F^app. How will this value depend on the shape and size of the paramecium (i.e. its mass (m) and effective radius (R))? How will it depend on its speed (v_T)?

d.  Solve the equation that you found in (c) for v_T. How does v_T depend on the applied force? 

e.  Suppose the paramecium is starting from rest and starts to move, and comes fairly quickly to its constant velocity, v_T. Describe how the three terms in the equation behave, illustrating your discussion with hand sketched graphs of x, v, a, F ^net, F^app, and F^viscous.

f. Now, return to the original question. Are the perspectives of Liz and Jack contradictory? Consistent? If they are consistent, explain how they can be reconciled. If they are contradictory explain the inconsistencies. (Regardless of the position you take, please address both the argument of Liz and Jack.)