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+ Contact Glenn          The study of rockets is an excellent way for students to learn the basics of forces and the response of an object to external forces. The easiest rocket to build and fly is the compressed air rocket. The system uses an air pump to launch the rocket and the rocket coasts throughout the rest of the flight. Air rockets have no engine to produce continuous 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 on this page we present the equations that describe ballistic flight.

Strictly speaking, 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, air rockets generate a moderate amount of aerodynamic drag and are not strictly ballistic. Ballistic flight is, however, a good first approximation to the flight of an air rocket. The 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.

To simplify our analysis, we assume a perfectly vertical launch. If the launch is inclined at some angle, we can resolve the initial velocity into a vertical and horizontal component. The horizontal motion is uniform because there is no external force in the horizontal direction. Weight is the only force acting on the object and weight is always vertical. 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 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 Java 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 RocketModeler III simulation program.

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