Unicellular organisms such as bacteria and protists are small objects that live in dense fluids. As a result, the resistive force they feel is large and viscous. Since their masses are small their motion looks very different from motion in a medium with little resistance.  In this problem we'll model the motion of a paramecium on a spreadsheet and explore how these situations differ.   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.  Since in this problem we are exploring how the motion of the paramecium depends on the parameters of applied force, mass, and resistance, we wont worry about how the cilia move to produce a consistent forward force.

1. Write the equation for Newton's second law for a paramecium feeling two forces: an applied force, F^app, and a viscous force. The viscous force takes the form F ^viscous = - 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.)
2. If the paramecium has a mass m and an effective radius R, what must be the applied force if the paramecium is moving at a constant velocity v_T? (Note: We have given you no numbers.  What is wanted here is an equation that could give a value for  F^app if we knew the values of m, R, and v_T.)
3. Solve the equation that you found in (b) for v_T.  How does v_T depend on the applied force? 
4.  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 full N2 equation behave, illustrating your discussion with hand sketched graphs of x, v, a, F ^net, F^app, and F^viscous.
5. 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.

Are the perspectives of these two students consistent or contradictory? 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.)