The study of the
flight of a baseball
and the "bend" of a
soccer kick
are excellent ways for students
to learn the basics of forces and
the response of an object to external forces.
A ball in flight has no engine to produce
thrust, so the
resulting flight is similar to the flight of shell
from a cannon, or a bullet from a gun. This type of
flight is called
ballistic flight
and assumes that
weight
is the only force acting on the ball.
In reality, a baseball or a soccer ball in flight generates a moderate amount of
aerodynamic drag
and is not strictly ballistic. On this page we develop the
equations which describe the motion of a flying ball
including the effects of drag.

At launch the ball is inclined at some angle to the vertical, so we
resolve
the initial velocity into a vertical and horizontal component.
Unlike the ballistic flight equations, the horizontal
equation includes the action of aerodynamic drag on
the ball.
We will first consider the vertical component and then develop the
equations for the horizontal component.

In the vertical plane, the only forces acting on the
ball are the forces of weight and drag.
There is a characteristic velocity
which appears in many of the equations that is called the
terminal velocity
because it is the constant velocity that the object sustains during
a coasting descent.
Terminal velocity is noted by the symbol Vt.

During the vertical descent, for a light object,
the weight and drag of an object are equal and opposite.
There is no net force acting on the ball and the vertical acceleration
is zero.

a = 0

W = D

where a is the acceleration,
W is the weight, and D is the drag.
The weight of any object is given by the
weight equation:

W = m * g

where m is the mass of the object and g is the
gravitational acceleration equal to 32.2 ft/sec^2 or 9.8 m/sec^2
on the surface of the Earth.
(The gravitational acceleration has different values on the
Moon and on Mars.)
The drag is given by the
drag equation:

D = .5 * Cd * r * A * Vt^2

where r is the
gas density,
Cd is the
drag coefficient which characterizes
the effects of shape of the ball,
A is the cross-sectional
area
of the ball, and Vt is the terminal velocity.

On the figure at the top, the density is expressed by the Greek symbol
"rho". The symbol looks like a script "p". This is the standard symbol used by
aeronautical engineers. We are using "r" in the text for ease of translation
by interpretive software.

The gas density has different surface values on the Earth and
on Mars and varies with altitude. On the Moon the gas density
is zero.
Combining the last three equations, we can determine the terminal
velocity:

m * g = .5 * Cd * r * A * Vt^2

Vt = sqrt ( (2 * m * g) / (Cd * r * A) )

Now, turning to the ascent trajectory, the ball is traveling
at an initial vertical velocity Vo.
With the positive
vertical coordinate denoted by y, the net vertical force Fnet
acting on the ball is given by:

Fnet = -W -D

Because the weight of the object is a constant, we can use the
simple form of Newton's second law to solve for the vertical
acceleration:

Fnet = m a = -W - D

m a = - (m * g) - (.5 * Cd * r * A * v^2)

a = -g - (Cd * r * A * v^2) / (2 * m)

Notice that the acceleration changes with time.
Multiply the last term by g/g and use the definition of the
terminal velocity to obtain:

a = -g * (1 + v^2 / Vt^2)

The acceleration is the time rate of change of velocity :

a = dv/dt = -g * (1 + v^2 / Vt^2)

Integrating this differential equation:

dv / (1 + v^2 / Vt^2) = -g dt

Vt * tan-1(v/Vt) = -g * t

where tan-1 is the inverse
tangent
function, and t is time..
The limits of integration for velocity v is from Vo to V
and the limits for time t is from 0 to t:

tan-1(V/Vt) - tan-1(Vo/Vt) = - g * t / Vt

tan-1(V/Vt) = tan-1(Vo/Vt) - g * t / Vt

Now take the tangent function of both sides of the equation
using the
trigonometric
identity:

tan(a - b) = (tan(a) - tan(b))/(1 + tan(a)*tan(b))

on the right hand side to obtain:

V/Vt = (Vo/Vt - tan(g * t / Vt)) / (1 + (Vo/Vt) * tan (g * t / Vt))

V/Vt = (Vo - Vt * tan(g * t / Vt)) / (Vt + Vo * tan (g * t / Vt))

This is the equation for the velocity at any time during the vertical ascent.
At the top of the trajectory, the velocity is zero. We can solve the velocity
equation to determine the time when this occurs:

Vo/Vt = tan(g * t(v=o) / Vt)

t(v=o) = (Vt / g) * tan-1(Vo/Vt)

To determine the vertical location during the ascent, we have to use
another identity from differential calculus:

dv/dt = dv/dy * dy/dt

dv/dt = v * dv/dy

We previously determined that

dv/dt = -g * (1 + v^2 / Vt^2)

v * dv/dy = -g * (1 + v^2 / Vt^2)

(v /(1 + v^2 / Vt^2)) * dv = -g dy

Integrating both sides:

(Vt^2 / 2) * (ln (v^2 + Vt^2)) = - g * y

where ln is the natural logarithmic function.
The limits of integration for velocity v is from Vo to V
and the limits for direction y is from 0 to y:

(Vt^2 / 2) * (ln (V^2 + Vt^2) - ln (Vo^2 + Vt^2) = - g * y

Notice that the location equation is pretty messy! For a given time t,
we would have to find the local velocity V, and then plug that
value into the location equation to get the location y.
At the maximum height ymax, the velocity is equal to zero:

ymax = (Vt^2 / (2 * g)) * ln ((Vo^2 + Vt^2)/Vt^2)

Here's a Java calculator which solves the vertical
equations presented on this page:

Ballistic Flight Calculator

Select planet, mode, and units

If drag included, enter weight or mass,

area, drag coefficient, altitude, or air density

Press Compute Button

Input

Output

Terminal Velocity

Max Height

Time to Max Height

To operate the calculator, you first select the planet using the choice button
at the top left.
For purely
ballistic flight,
select the "Ignore Drag" option with the middle choice button.
For flight with drag, select "Include Drag" with the middle choice button.
You can perform the calculations in English (Imperial) or metric units.
Enter the initial velocity.
Since we are performing the calculation with drag, we must specify
the object's weight,
cross sectional area,
and a
drag coefficient. The air density is
determined
by the altitude, or it can be input directly.
Press the red "Compute" button to compute the maximum height and
the time to maximum height.
The program also outputs the
terminal velocity
as described above.

We provide an on-line web page that contains only this
calculator.
You can also download your own copy of the calculator for use off-line.
The program
is provided as Fltcalc.zip. You must save this file on your hard drive
and "Extract" the necessary files from Fltcalc.zip. Click on "Fltcalc.html"
to launch your browser and load the program.

Notice If you toggle the middle choice button between "Ignore Drag"
and "Include Drag", you will notice that the computed height is always
less when including the drag. The amount of the difference indicates the
importance of drag for certain flight conditions. Also consult the
terminal velocity
web page for some warnings concerning cases with high terminal velocity.
If you hold the initial velocity constant, and increase only the weight,
you will notice that the maximum height gradually approaches the
ballistic flight value.

The horizontal equations are a little easier, since the only net force
acting on the ball is the drag:

Fnet = m a = - D

m a = - (.5 * Cd * r * A * u^2)

a = - (Cd * r * A * u^2) / (2 * m)

where u is the horizontal velocity. We can use the terminal
velocity to simplify this equation:

a = du / dt = - g * u^2 / Vt^2

(1 / u^2) du = - (g / Vt^2) dt

Integrating the equations, with the limits on the velocity from the
intial velocity Uo to U, we obtain:

u = dx/dt = Vt^2 * Uo / (Vt^2 + g * Uo * t)

The horizontal velocity is inversely dependent on the time. We can
similarly solve for the location x at any time by integrating
the velocity equation:

x = (Vt^2 / g) * ln( (Vt^2 + g * Uo * t) / Vt^2 )

You can study the aerodynamic effects on a falling
object with the
DropSim
simulation available at this web site.
You can also study the flight characteristics of a ball
with drag by using the on-line
HitModeler
simulation program for a hit baseball, or the
SoccerNASA
simulation program for a kicked soccer ball.