Mathematics in Physics — Free Body Diagrams and Forces in a Rocket Launch Project

In my AP Physics class, we were tasked with launching a rocket to as high as we could using principles of forces and kinematics. Of course, physics is the natural science to which mathematics is the most influential, so this was an extremely exciting project for me to complete. The project was separated into two phases — one where the goal was simply to launch the rocket a good height, around 80 feet, and one where we needed to launch the rocket as high as we could with an egg inside and ensure the egg survives due to a parachute. My team and I used many materials including a trash bag, soda bottles, newspaper, tape, strings, and plastic sheets to construct our rocket and parachute.

Now here comes the fun part — the math involved in the rocket launch. After constructing a free body diagram of the rocket in the first phase, it’s evident that there is an upward force of thrust that outweighs the downward forces due to gravity and wind resistance, and on the way down, the downward force of gravity is stronger than the upward force of air resistance.

After using video analysis and motion sensing in Vernier Analysis through LoggerPro, we get some interesting data — the maximum height was 81.66 feet or 24.9 meters, a thrust acceleration of 127.1 m/s², and a maximum velocity of 32.8 m/s². We have achieved the necessary requirements for phase 1.

Now, onto phase 2. The forces acting on the rocket are the same, but the air resistance has a greater magnitude due to the now existing parachute on the way down.

After using LoggerPro, we get the following data for our launch: a maximum height of 71.97 ft or 21.9m, a thrust acceleration of 118.2 m/s², a maximum velocity of 23.7 m/s, and a downward acceleration of 4.29 m/s. It was time to use mathematics to determine the force values acting on our rocket.

Thrust is simple — due to Newton’s Second Law, F = ma, and so the force of thrust equals the mass of the rocket, being 0.85kg, multiplied by the thrust acceleration, which we got from LoggerPro as 118.2 m/s². 0.85 * 118.2 = 100.45, so the force of thrust propelling the rocket upward in our phase 2 launch was 100.45N. Weight, or force due to gravity, is also simple — F = ma, and the acceleration in weight is obviously the acceleration caused by gravity, being 9.8 m/s², so F = 9.8m = 0.85 * 9.8 = 8.33, so the rocket’s weight is 8.33N. Before determining the upwards net force, solving for the force due to air resistance was necessary, but luckily, we can refer to our free body diagram so sum up a few forces. On the way down, the net force is equal to the weight minus the air resistance since the weight has a greater magnitude, so F = ma = mg — AR. We luckily know the net acceleration on the way down due to LoggerPro, so substituting values gets us 0.85 * 4.29 = 0.85 * 9.8 — AR, and after some simple calculations and algebra, AR = 4.686, so the force due to air resistance is 4.686N. Finally, it is time to determine the upwards net force on the rocket, being the force due to thrust minus both weight and the force due to air resistance. F = ma = T — mg — AR = 100.45–8.33–4.686 = 87.249, so the upwards net force acting on the rocket is 87.249N.

Although the entire project was fun, I especially enjoyed constructing the free body diagrams and completing the force calculations for the rocket, as it is purely mathematics and logic involved in doing so. Our rocket launches were successful in both phases 1 and 2, with the egg surviving in phase 2 and the parachute deploying. I look forward to broadening the scope of mathematical physics in the future, especially in the current unit we are undergoing: circular motion and universal gravitation!