Physics 160

The problem asked that you consider and compare the physical behavior of a ball being thrown straight upwards into the air, with some non-zero initial velocity, and to consider (at least) two different cases:

• one with air drag, where we modelled the air drag as a velocity-dependent force in a reasonably standard way, namely dv/dt = a = a(v) = -[g + b v |v|] . for constant parameters, g and b,
• and one with the standard approach without air drag, which you can obtain by simply setting b=0 in the above approach, or just use the results from Chapter 2 for simple motion under constant acceleration.

I suggested reasonable initial conditions for you to use, although you were also certainly free to use additional ones if you like:

1. the initial (upward) velocity, v₀ = 20 m/s,
2. and the parameters b = 1/(40 m) = 0.025 /m ,
3. as well as the (usual) gravitational constant, g = 9.8 m/s².

The problem is then to first calculate the velocities and the trajectories, and then to analyze the numbers so created, and to create graphs versus time.
There are, at least, two reasonable aspects to analyzing the numbers. The first aspect is to make graphs showing the behavior, separately for the height, h = h(t), and for the velocity, v = v(t). One should show the air-drag case and the no-air-drag case together, so that they may easily be compared.
Examples were given in the problem statement, with the input parameters discussed above. They were created with the use of the algebraic-computation program Maple.
The curves shown below were computed with the spreadsheet program Excel.

Plots of vertical displacement versus time. The curves are stopped when it hits the ground.

Plots of vertical velocity versus time. The curves continue even after it hits the ground, to show the onset of the terminal velocity.

These graphs make it clear, as expected, that air drag both reduces the maximum height to which the ball may go and the maximum speed that it attains when it returns to the ground.
As well that speed on return is less than the speed with which the trip began.
Also we see that the trip upward and also the trip downward take less time than the trip without air drag, although the trip downward now takes more time than the trip upward.

For the air-drag case, and the gravity-only case, as graphed above, I now list numerical values that support the conclusions above.
The two cases both begin with an initial velocity, upwards, of 20 m/sec.
I then gleaned the rest of the following numbers from the given (Excel) data in the following ways, where all the numerical values given are in SI units:

1. for the initial acceleration, just insert the values of g, v₀, b, and n into the formula for the acceleration;
2. for the time at the top of the trajectory, we know that the velocity at the top is zero, so that we simply search down the list of computed velocities until we find that calculated velocity that is nearest to zero, and choose the time corresponding to that particular calculation;
3. for the height at the top of the trajectory, having already found the time, we read off the corresponding value for x;
4. to determine the time for the return trip we know that the return occurs when the height is zero, i.e., when x=0, again, so that we now scan further down the list of calculations until we find that calculated value of x that is nearest to 0, and read off the associated time, and then subtract the time for the trip upward, already determined, to obtain the time downwards;
5. to determine the speed on impact we simply read across, for the time of impact just determined, to the corresponding values for v;
6. I used a time interval, t, of 0.025 seconds.

It is worth noting that the gravity-only case numbers, above, are taken from the Excel spreadsheet; if, instead, one uses the exact numbers calculated from the formulae in Chapter 2, then we obtain t_top = 2.041 s, and h_top = 20.41 m.
This points out the approximate nature of the numerical calculations!

As an addendum to the actual list above of quantities requested for a complete solution, and assuming that all who tried this problem used Excel as a calculational tool---as I expected---I note here that the Excel directions already given are pretty much self-sufficient. Nonetheless, I here append the top several rows of the Excel spreadsheet, and also a few comments below to further help explain it.
As shown in the webpage on the use of Excel for such purposes, I have put in the parameters needed in the following places:

• dt in the Excel cell labelled E8,
• g in the cell G11,
• b in F8,
• n in G8, [In the program I kept the parameter n, but just set it to 1.]
• v₀ in K10,
• x₀ in J10,

• and the generic, continuing values for t in column I,
• the generic values for x in column J, beginning, as above, with t=0 in row 10,
• the generic values for v in column K, beginning with t=0 in row 10,
• the generic values for a in column L,
• and
  • the half-step values for x in column O,
  • the half-step values for v in column P,
  • the half-step values for a in column Q,

then we may write out the generic formula for the acceleration as follows, giving it, say, as it would have appeared in row 12, a few steps down in the calculation:

1. the half-step value for a, put in column Q, as -($G$11 + $F$8*K12*(ABS(K12))^($G$8)),
2. and the full-step value for a, put in column L, as -($G$11 +$F$8*O12*(ABS(O12))^($G$8)),

If, instead, you did the problem via integration, I append a Maple worksheet which solves the differential equation and displays the results both as equations and in graphical form.
The equations that define the solution are the following, which involve

• a parameter A = [gb]^1/2, with units of 1/sec , which provides a scale, and
• a parameter d = (1/A)arctan[bv₀/A] , determined by the initial velocity, v₀:

There are of course other methods to integrate the equations, including just sitting down and doing it; however, I find this a convenient tool to do such complicated integrations.

It might also be worth pointing out that the complete integration cannot be done in closed form, in terms of known functions, for cases other than when the air drag is modelled by the first or second powers of the velocity.