Littlewood's Law by Alberto Gassol

 

Every rocketeer wants to know how high his rocket flew. In today's technology there are relatively inexpensive electronic processors that can register altitude with great precision. The traditional method has been trigonometric triangulation but, I want to tell you about a third method which can be used to measure a rockets peak altitude easily and with good precision, it is called Littlewood's law.


figure 1

If we fly a rocket ballistic as in figure 1 and we register the total flight time T, then the peak altitude or apogee obtained by the rocket is:

Sounds simple? the only thing we needed is a chronometer or better still a camcorder with which to register the flight, count the time, multiply this number by itself and by 1.23 thus obtaining the height reached by the vehicle in metric units.

This method is based on the surprising fact that if a ballistic flight without friction and a real flight with friction have both the same total time of flight, then both will almost have the same height at apogee (figure 2)


figure 2

The correct expression of equation 1 is:

This simple relation is known as Littlewood's Law named for mathematician J.E. Littlewood, who did some artillery-related math for the British military during World War I.

If you have any doubts on the precision of this method, compare its predictions to those of the simulator of your preference or, to your own in-the-field measurements, you may be surprised how well it does over a wide range of conditions.

I will use as an example the rocket I flew for "Cohetes de America", an international event summoned by ACEMA on October 2, 2004. According to my simulator of preference "Launch", the total time of flight was T = 40 second and reached a height of h = 1.830 meters (figure 3). Using Littlewood's Law we have:


less than 8% of a difference.


figure 3 – LAUNCH simulator

in summary:

  1. In the case of a ballistic flight all that is needed to calculate peak altitude is total flight time. If we only obtain time at apogee due to parachute deployment, then apogee time times two is the total time used in the calculation.
  2. It is good practice that the observer registering the time be at least the same distance from the launch pad as that expected for peak altitude.
  3. A smoke trail is highly recommended to ease flight tracking.

 

Deduction of equation 1

Assume an ideal non friction flight whose vertical coordinate equations of motion are:

 

References:

  • Dean’s Benchtop – Water rocket flight equations http://www.et.byu.edu/~wheeler/benchtop/flight.php#graphical
  • Hyperphysichs: http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html#tra5
  • Program "Launch" DARK: http://www.dsri.dk/%7Ejrp/Launch22A.zip
  • ACEMA (Asociación de Cohetería Experimental y Modelista Argentina): http://acema.arg54.com.ar/index.htm