Flying model rockets is a relatively
safe
and inexpensive way for students
to learn the basics of forces and
the response of vehicles to external forces.
Students can also use math techniques learned in high school to determine
the performance of the rocket during the
flight.

On this slide we show a simple way to determine the maximum altitude
the rocket reaches during the flight.
The procedure requires two observers and a tool like the one shown
in the upper right portion of the figure to measure angles. The observers
are placed some distance L apart along a reference line which is shown in white
on the figure. You can lay a string of known length along the ground between the
observers. As the rocket passes its maximum altitude, observer #1 calls out
"Take Data", and measures the angle a between the ground and the rocket.
This measurement is taken perpendicular to the ground. Observer #1 then measures
the angle b between the rocket and the reference line.
This measurement is taken parallel to the ground
and can be done by the observer facing the rocket, holding position, and
measuring from the direction the observer is facing to the reference line
on the ground. When the second observer hears the call, "Take Data", the
observer must face the rocket and measure the angle d from the ground to the
rocket. The second observer must then measure the angle c, parallel to
the ground, between the direction the observer is facing and the reference
line in the same manner as the first observer.
Angles a and d are measured in a plane that is perpendicular to the
ground while angles b and c are measured in a plane parallel to the ground.

With the four measured angles and the measured distance between the
observers, we can use some relations from
trigonometry
to
derive
an equation for
the altitude h of the rocket. The equation is

h = (L * tan a * tan d) / ( cos b * tan d + cos c * tan a)

h = (L * tan b * tan d) / (cos c * (tan c + tan b))

An alternative equation, which is equivalent to the previous equation,
is also shown on the figure:

h = (L * tan d * sin b) / sin(b + c)

where the sine (sin) is another trigonometric function. Notice that
in this equation you have to add the angles b and c before
evaluating the sine in the denominator. This is called a "double angle" formula.

If we eliminate angle d, the resulting equation is:

h = (L * tan a * tan c) / (cos b * (tan b + tan c))

An alternative equation, which is equivalent to the previous equation,
is also shown on the figure:

h = (L * tan a * sin c) / sin(b + c)

You can use any of these equations to determine the height of any object
from a tall tree to a flying rocket.
If you have your observers take all four angle measurements, you can actually
make three calculations of the height, which can help to eliminate errors
in the measurments.
If you do not know trigonometry, you can still determine the altitude
of the rocket by using a
graphical solution
from the four angle measurements.

As a further check, here's a JavaScript calculator which will solve the
equations presented on
this page.

Altitude Calculator

Enter values for angles and reference length

Choose compute mode

Press compute button

Input

degrees

degrees

degrees

degrees

Output

Altitude

You can enter the four angles shown in the graphic in the white boxes
labeled "Angle A" through "Angle D". You then choose the mode for the
calculation using the drop down menu. Mode #1 uses all four measured
angles, Mode #2 ignores angle a, and Mode #3 ignores angle d.
You perform the calculations by pushing the red "Compute" button. You
should compare all three calculated results, your own hand calculations,
and a graphical calculation to determine and minimize errors in the
measurements.
Calculations and input can be entered in either
English or Metric units by using the "Units" choice button.

You can download your own copy of this calculator for use off line. The program
is provided as Altcalc.zip. You must save this file on your hard drive
and "Extract" the necessary files from Altcalc.zip. Click on "Altcalc.html"
to launch your browser and load the program.