An excellent way for students to gain a feel for
aerodynamic forces
is to fly a
kite.
Students can also use math techniques learned in high school to determine
the altitude of the kite during the
flight.

On this page we show a simple way to determine the altitude of a flying kite.
The procedure requires an observer in addition to the kite flyer,
and a tool, like the one shown
in the upper portion of the figure, to measure angles. The observer
is stationed some distance L from the flyer 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 flyer and the observer to make this reference line. A long line
will produce more accurate results.
To determine the altitude, the flyer calls out
"Take Data", and measures the angle a between the ground and the location
of the kite.
This measurement is taken perpendicular to the ground. The flyer then measures
the angle b between the kite and the reference line.
This measurement is taken parallel to the ground
and can be done by the observer facing the kite, holding position, and
measuring from the direction the observer is facing to the reference line
on the ground. When the observer hears the call, "Take Data", the
observer must face the kite and measure the angle d from the ground to the
kite. The 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 flyer.
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 flyer and observer 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 kite by using a
graphical solution
from the four angle measurements.

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

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.