A Sounding rocket is a sub-orbital unguided science-oriented payload rocket. Being unguided, it is very important to track its trajectory, and predict it long before launching. A rocket that flew into Fairbanks would not do very well!
In the very simplest form, a sounding rocket can be thought of as a particle, and as such would be subject to constant acceleration from the Earth, and no drag or other forces, except its thrust. For many situations this model is good enough! The model does however, make many assumptions (by eliminating all of the other forces acting on a rocket.) It also assumes that the mass of a rocket remains constant throughout flight. This is of course not true. Propellant is ejected to propel the rocket forward.
However, if we are to use this model, it can be characterized by the basic kinematic equations of physics:
First the range of a given rocket launched at an angle f from the horizon is:
Where range is the rockets range in kilometers, v0 is the initial velocity in meters per second, and g is the gravitational acceleration of earth. Thus when the angle f is 90°, the horizontal range of the rocket is 0.
The altitude the rocket will reach is as follows:
Where t is the time of flight in seconds. To find the maximum altitude, we simply modify the range equation to reflect the "y" component of the vehicles trajectory. We then get that the maximum altitude occurs when the vertical velocity of the vehicle is 0, so:
Where the parameters are all the same as the first (range) equation.
This provides a basic and simplified, yet powerful view of the basic trajectory of a ballistic sounding rocket.