Physics 160

I want to consider in more detail the notion of the drag force on objects moving through air, as briefly described in Section 6-3 of your text.
When any object moves through air, it encounters a force we could call air resistance, or air drag. This force is rather different than many of the forces we have been considering so far, since its magnitude and direction depend on the magnitude and direction of the velocity of the object.

• The direction of the drag force is easily seen to be always opposite to the direction of the velocity.
• The magnitude of the drag force depends on many things. A reasonable approximation for it is that it is proportional to some power of the velocity, either
  1. depending on the first power, i.e.,
     F_drag = -kv , where k is a constant independent of the velocity but depending on the shape and mass of the object,
  2. or, depending on the second power,
     F_drag = -bv|v| , where b is again a constant depending on the shape and the mass of the object, and we insert the magnitude of the velocity so that we do have a dependence on the second power, but a sign so that it always opposes the motion.

I want you to create a set of values for both velocity and height versus time, from the time that the ball is initially thrown up until it comes back to the ground, and to also compare them to the case without drag, which of course one does by using formulae from Chapter 2.
After you have the numbers, then please make graphs illustrating those numbers. Examples of such graphs are appended below.
You are welcome to use the values for the parameters that I have chosen above, namely b=1/(40 m) and v0 = 20 m/s, or you should feel free to use any other reasonable choices that you might prefer, or both.
You must turn in the the lists of values that you have calculated, a brief explanation of the method you used to calculate them, and the graphs illustrating those values.

Various methods are available to do this problem, but you should note that the acceleration is NOT constant; therefore, you may NOT use those formulae from Chapter 2 which are for constant acceleration.
Although all I ask is that you do this for

• For example, you may use a numerical approach, where you first start with t=0, and v=v0, calculate a, and then use a very small time-step, t, to calculate a new v, using the standard approximation to the derivative, namely v = a t, so that when t = t, then v = v0 + v. At that point you have a new v, so you repeat the process, i.e., calculate a new a, and then a new v, and a new v, etc. Eventually this gives you a series of values for v as a function of time, and one may make a graph.
  Once one has all these v's and t's, they may be numerically integrated again, via the same procedure as above, but simpler, to find x as a function of time.
  Note that Excel is a program that is well adapted to doing these sorts of numerical integrations. As well, some hand calculators can be taught to do this. CAPS had recently a workshop on such an approach.
• A different approach is to use the calculus---somewhat more advanced calculus than we have so far needed in class---to integrate the equation above for v.
• Yet another approach is to use some computer-based software that does integrals for one. Examples are Maple and Mathematica.

Some (old) helpful hints may be found at this link.

The two graphs shown below are

• a graph of velocity versus time, for the constants described above, with air drag, and compared to zero air drag:

  

  You can see that the one with drag loses speed more quickly, comes to the top---where v=0---sooner. As well, however, you can see the terminal velocity coming, where that curve more or less levels out.

• and a graph of height versus time, with and without air drag.

  

  You can see that the one with drag rises more slowly, and less high, than the one without drag; as well, it returns to the earth sooner than the one without drag.