Analysis of Baseball Motion Including Air Resistance
Fenway Problem Part 2
When we assumed that there was no air resistance opposing the motion of the baseball,
we could write relatively simple equations describing the motion of the ball.
vx = v0x
vy = v0y – gt
x = v0xt
y = y0 + v0yt – ½ gt2
Air resistance is a force acting on the ball in addition to the force of gravity. Furthermore,
that force is not restricted to just the vertical component of velocity. As does the
gravitational force, the force of air resistance causes the velocity to change.
It turns out that air resistance is not constant. It depends on the speed of the ball: the
faster the ball goes, the greater the resistance.
The formula for the resistance force is Fair = ½ Cd A v2, where Cd is a drag coefficient,
 is the density of air, A is the cross-sectional area of the baseball, and v is the speed.
Since F=ma (Newton’s second law of motion), we can substitute ma for Fair. Since m and
all the coefficients are constants, we can finally calculate a = kv2, where a is the
acceleration (really deceleration) effect of air.
The equations for velocity of the ball become:
vx = v0x – at
 vx = v0x – kvx2 t
vy = v0y – gt – at  vy = v0y – gt – kvy2 t
Ordinarily, the left side of an equation is the dependent variable, the right is the
independent variable. But in this case, the dependent variable, velocity, is present in both
the left and the right side of the equation. That is, the velocity is dependent on the
velocity. There is an additional difficulty in that the use of the initial velocity is no longer
legitimate.
We can get around these mathematical difficulty using spreadsheets if we remember that
a spreadsheet can use previous values to calculate new values. We used this principle in
setting up the time column when we said that the present time is the previous time plus
some constant increment in time. tpresent = tprevioust + t. In code, we said “Take the value
of time in the cell above the present cell and add an increment of time, e.g. 0.2 seconds to
it. Put the result in the present cell.” We might have written in A6 the code =a5+0.2.
In this case we can say “Take the previous velocity. Use it to calculate the term kv2 in the
equation for the present velocity.” In doing this we assume that the velocity during that
interval is constant. If we take the increment of time to be very short, then the velocity
during that time interval can be assumed to be constant with little error. (This assumption
is implicit in the calculation of k above.) This procedure is illustrated graphically for vx
below. Notice that the initial velocity has been replaced by the previous velocity and that
the time is the time interval.
Previous velocity
Present velocity with air resistance effect subtracted from the
previous velocity
vx = vx – kvx2 t
Present
velocity
Previous
velocity
t
We can now write the spreadsheet equations as:
vx present = vx previous – kv2x previous t
vy present = vy previous – gt – kv2y presvious t
where t is the time interval.
Since we are assuming constant velocity over the time interval, the calculation of distance
becomes simple. We know that for constant velocity d = vt. Therefore, once we have the
value of the velocity for a time interval, x and y can be calculated as the previous distance
plus the change in distance, where e.g. x = vx previous t
xpresent = xprevious + vx previous t
ypresent = yprevious + vy previous t
Notice that we are adding up the areas under the velocity versus time curve; i.e. distance
is the area under the velocity versus time curve, as discussed in class.
Fenway Park Problem: Part 2
Solve the Fenway problem described in Part 1 but include the effects of air resistance.
Answer all the same questions.
Set up a spreadsheet as shown below, where theta is the initial angle, , delta is the time
interval, , and rho is the density of air, . The values are those for the Fenway problem.
In B5 and B6 enter the formulas to calculate the horizontal and vertical components of
the initial velocity, and use those values in the main part of the spreadsheet. In E6, enter
the formula for k, given below and used in calculating air drag. The spreadsheet is set up
so that you can change the given values for different initial conditions, types of balls, and
altitudes above sea level, where the density of air is less.
1
2
3
4
5
6
7
A
x0=
y0=
v0=
theta0=
v0x=
v0y=
time, s
B
0
1.0
35
43
vx,
m/s
C
vy,
m/s
D
delta t=
Cd =
rho=
A=
m=
k=
x, m
E
0.1
0.5
1.2
.004158
0.142
y, m
F
vxair, m/s
G
H
I
vyair, m/s
xair, m
yair, m
k = CdAm
.
That way the spreadsheet will be useful to find out what might happen if the stadium is in
Denver where the density of air is less, or even so you could use the spreadsheet to
characterize the trajectory of a basketball or a golf ball.
Set up the spreadsheet using v0x, v0y and k in the velocity equations.
In column A, enter the times, using t as the time increment. These times are for plotting
only, since the absolute time is not used in the formulas, but t is.
In column B, use the iterative procedure outlined above to calculate the distance the ball
travels in the x direction but without the air resistance term. v0x should be in the cell
corresponding to t=0. In the next and subsequent cells will be vx present = vx previous because
without air resistance the initial velocity in the x direction does not change, i.e. the air
resistance term is not used in the formula.
In column C, enter the iterative procedure formula for vy but without the air resistance
term. In columns D and E, enter the iterative procedure formulas for x and y without the
air resistance term.
Compare the results to the results you got using the algebraic formulas in Fenway
Problem: Part 1. Are they the same or nearly the same?
Next, in columns F, G, H, and I calculate vx, vy, x, and y using the iterative procedure and
including the air resistance term as described above.
Then, for no air resistance, plot x as the horizontal variable and y as the vertical variable
(columns D and E). Make the maximum value on the x axis 130 meters. Set the x and
y axis scales so that no negative values are plotted. Answer the Fenway problem
questions and compare the results to those found in the Fenway Problem Part 1.
Then plot x as the horizontal variable and y as the vertical variable including the air
resistance term (Columns H and I). Make the maximum value on the x axis 130
meters. Set the x and y axis scales so that no negative values are plotted. Compare the
results when air resistance is included to those when air resistance is neglected.
Was neglecting air resistance a good assumption?
What is the best fit second order polynomial equation for the y versus x curve when air
resistance is neglected? What is the best fit second order polynomial equation when air
resistance in included. For the air resistance trajectory, does a third order polynomial give
a better fit?