Altitude by Trigonometry

"How high did it go?" The basic question we are going to ask after every flight (along with "Did you see which tree it landed in?") Since we can't measure the altitude directly, we are going to take advantage of some right triangle trigonometry to help us through this problem. Let's review some basics.

We label a right triangle (defined as a triangle with one angle of 90o) as shown in the diagram to the right. To review some basics: The side opposite the right angle is the longest side and is called the hypotenuse. The
horizontal side is the base, and the vertical side will be our height or altitude. There is a relationship between these sides in a right triangle that is given by the Pythagorean Theorem: The sum of the squares of the base and the altitude is equal to the square of the hypotenuse, or put mathematically, a2 + b2 = c2. Thus, given two sides, we can easily find the third.
Not that this is going to help us in our problem. At best, we can measure the base of our triangle, or the distance from our observer to the launch pad. That only gives us one side - how are we going to get another?

 

There are a series of ratios that define the lengths of sides based upon the angles of the triangle. These are called Trigonometric Functions and they are known and constant for all right triangles. The three most common of these, Sine, Cosine and Tangent will be useful to us. Again, consider the triangle shown and these definitions:

Sine (abbreviated Sin) - The sine is defined as the ratio of the opposite side (of the angle) to the hypotenuse. The sine has a range of 0 at an angle of 0o to 1 at an angle of 90o. The opposite side of the triangle is our altitude, although we probably won't use the sine function very often.

Cosine (abbreviated Cos) - The cosine is defined as the ratio of the adjacent side (of the angle) to the hypotenuse. The adjacent side of our triangle is our base, or our measured distance from the observer to the launch pad. The cosine has values in the range of 1 for 0o to 0 at an angle of 90o. At an angle of 45o, the sine and the cosine are equal.

Tangent (abbreviated Tan) - The tangent is defined as the ratio of the opposite side of the triangle to the adjacent side, or the ratio of the altitude to the base. The tangent can range in value from 0 at 0o, to infinity at 90o. (The value of the tangent at 90ois undefined). Actually, this can be a problem for us as errors at really steep angles will be magnified. We need to make sure we keep our base as long as possible. The tangent function will be our function of choice since we will know our base length and will only need to determine the angle from the observer to the rocket to solve for altitude.  

In a nutshell:

 

Finding the altitude will be a relatively straight forward calculation. We will measure the distance from the observer to the base of the launch pad. Let us suppose that is 50 meters. We will use a device called an astrolabe to measure the angle of inclination. (A link to building a home-made astrolabe can be found at http://cse.ssl.berkeley.edu/AtHomeAstronomy/activity_07.html)
One individual will sight the rocket through the tube on the top while a second individual will read the angle on the side and record the highest angle reached. Then we will be able to compute our altitude using the angle and the measured distance from the observer to the launch pad. Here is how:

solving for the altitude,

Let us suppose our measured angle is 75o. The tangent of 75o (obtained from a table or a calculator) is 3.732.
Thus, our altitude is 50 meters x 3.732 = 187 meters. 
One of the problems with this method is that it assumes that the rocket travels straight up. When there is wind, the rocket will no longer be directly above the launch pad, and our computed altitude will not be correct. A way to help overcome this is to take two angle measurements from directions that are at right angles to each other with respect to the launch pad. This will help us get a closer approximation of the true altitude the rocket reaches.

Finally, don't forget to look for inaccuracies where they exist. For example, how accurately can we read the angle off of our astrolabe? How tall is the observer? Does this matter? Were we able to spot the apogee of the rocket?

Want to read more? Try:

http://aleph0.clarku.edu/~djoyce/java/trig/

http://www.ping.be/~ping1339/gonio.htm