The study of
model rockets, the
flight of a baseball,
or 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 continuous
thrust and 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 on this
page we present the equations that describe ballistic flight.
Ballistic flight only occurs under the ideal
conditions that
weight
is the only force acting on the object.
There is no
thrust
and no
aerodynamic drag
acting on an object in ballistic flight.
Such flight conditions would occur on the
Moon, where there is no
atmosphere to produce drag.

On Earth
a baseball or a soccer ball generate a moderate amount of
aerodynamic drag
and the flight path is not strictly ballistic.
Ballistic flight is, however, a
first approximation to the flight of a ball. The actual
flight equations
including drag are much more complex because the drag is constantly
changing throughout the flight. Drag depends on the
square of the velocity
and the velocity changes during the flight.
The drag also depends on the
air density
which is a function of the weather conditions and altitude.

For ballistic flight,
the ball is normally inclined at some angle
to the vertical (or horizontal) as it is launched. We
resolve
the initial velocity into a vertical component V0 and a
horizontal component U0.
The horizontal motion is uniform because there is no external
force in the horizontal direction. So, according to Newton's
first law
of motion, the horizontal velocity remains a constant and the
distance x which the ball travels is given by the velocity
times the expended time t:

U = U0

x = U0 * t

In the vertical plane,
weight is the only external force
acting on the object.
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 motion:

-W = F = m a = m dV/dt

where W is the weight, m is the mass, V is
the vertical velocity, t is the time, a is the acceleration,
and F is the net external force.
The positive direction is upwards, so the weight is preceded by a
negative sign. Solving the equation:

dV/dt = - W/m = -g

V = Vo - g t

where g is the gravitational acceleration which is equal to 32.2 ft/sec^2
or 9.8 m/sec^2 on the surface of the Earth.
The value of the gravitational acceleration is different on the
Moon and Mars.
Vo is the initial velocity leaving the
launcher. The location at any time is found by integrating the
velocity equation:

dy/dt = V = Vo - g t

y = Vo t - .5 g t^2

where y is the vertical coordinate.
With this general description of the motion of a ballistic object,
we can derive some interesting conclusions.

Notice that the flight equation includes no information about the object's
size, shape, or mass. All objects fly the same in purely ballistic flight.
This is similar to
Galileo's principle
that all objects fall at the same rate in a vacuum. If drag can be ignored,
the flight of the object depends only on the initial velocity and the
gravitational acceleration.

At the highest point
in the flight, the vertical velocity is zero. From the velocity
equation we can determine the time at which this happens:

V = 0

t = Vo / g

The time to maximum altitude varies linearly with the launch velocity.
Plugging this time into the altitude equation we obtain:

y = Vo (Vo / g) - .5 g (Vo / g)^2

y = .5 * Vo ^2 / g

The maximum altitude changes as the square of the launch velocity. Doubling
the launch velocity produces four times the maximum altitude.

Here's a JavaScript calculator which will solve the
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 ballistic flight, select the "Ignore Drag" option
with the middle choice button. On another page
we develop the equations for
flight with drag.
You can perform the calculations in English (Imperial) or metric units.
Enter the initial velocity,
then press the red "Compute" button to compute the maximum height and
the time to maximum height.
Notice that entering a different value for the weight or the
area does not change the computed maximum height.

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.

Now consider the impact with the ground at the end of the flight. At
impact the altitude is zero. Using the altitude equation:

y = 0

Vo t = .5 g t^2

t = 2 Vo / g

The total flight time varies linearly with the launch velocity. The
total flight time is twice the time to reach maximum altitude. So a
ballistic shell takes as long coming down as it does going up. If we
substitute this time into the velocity equation:

V = Vo - g (2 Vo / g)

V = - Vo

The velocity at impact has the same magnitude but opposite direction
as the velocity at launch.

You can study ballistic flight characteristics by using the
HitModeler
simulation program for the flight of a baseball.
Or you can use the
RocketModeler
program to simulates the flight of a model rocket.
Or you can use the
SoccerNASA
simulation program for the flight of a soccer ball.