By Anuraag Polisetty COSMOS Cluster 3 Dynamics of Rockets By Anuraag Polisetty COSMOS Cluster 3

Archimedes and Leonardo da Vinci Scientists Archimedes and Leonardo da Vinci

Archimedes Discoveries/Inventions Ancient Greek mathematician, physicist, engineer Born 287 BC in Syracuse, Sicily Discoveries/Inventions Able to estimate value of pi using a circle inscribed in two hexagons Doubled sides until closest value found between 3 10 71 and 3 10 70 (3.1408-3.1429) Created a “death ray” out of hundreds of mirrors to focus sun’s energy Designed screw pumps, innovative pulleys, weapons

Archimedes’ Principle Famous anecdote of Archimedes figuring out crown’s impurity by calculating displacement in water Most likely untrue Found that a body immersed in a fluid experiences a buoyant force equal to weight of fluid it displaces More likely method of finding impurities in crown Fundamental law in fluid mechanics Ship hulls displace enough water to provide buoyant force

Leonardo da Vinci Leonardo da Vinci (Born April 15, 1452 – Died May 2, 1519) Italian polymath – incredibly diverse talents and interests Painter, architect, mathematician, engineer, astronomer, inventor, geologist, historian, writer, and many other things Mostly well known for his paintings, but also made contributions to math and engineering Tactician and engineer during war Obsessed with phenomenon of flight – designs for parachute, flying machines, giant crossbow, winged machines

What are Rockets? Launch vehicles that use propellant to create vertical thrust Uses: Space exploration Satellite launching Military weapons Types of Rocket Engines: Solid Fuel Liquid Fuel Ion thruster Solid fuel rockets such as the two on the space shuttle launcher are very powerful but also simple and reliable. However, once they are ignited, they cannot be controlled, throttled, or stopped. Liquid fuel rockets are advantageous because they can be controlled pumps and valves to regulate thrust. These are mostly used for the space shuttle orbiter so it can be maneuvered in space and smaller missiles such as the German V-2. Downside: don’t provide as much thrust and are far more complex. Ion thrusters are boosters for use in space. While they have very low thrust, they have a very high specific impulse, meaning they can achieve very high velocities using a minimum amount of fuel.

History of Rockets Hero of Alexandria used principle of thrust to create a spinning turbine (100 BC) First rockets powered by gunpowder in China (1200s) Johann Schmidlap builds multi-stage rocket(1591) Used for show or warfare, highly inaccurate As you can see, Hero’s turbine heats water and uses steam as propellant to provide the thrust. This was just used for show to the people of his city. The Chinese created the first true rockets using gunpowder in a tube and a bamboo stick to guide it. These rockets were only used a couple times in warfare to just scare the enemy, and mostly used for fireworks. These kinds of rockets were highly inaccurate, you pretty much had to point them in the general direction you want to go and hope for the best. Rockets stayed the same until 1591, when German Johann Schmidlap built a multi stage rocket. A larger casing was used to propel upward, and then a smaller rocket shot out and gained extremely high altitudes. Goddard and liquid fuel rocket

Advances in Rocketry Newton’s Laws (1687) set the scientific foundation for modern rocketry William Hale developed spin stabilization for increased accuracy (1844) Tsiolkovsky’s rocket equation derived from Newton’s 2nd Law: ∆𝑣=𝑣 𝑒 𝑙𝑛 𝑚 0 𝑚 1 Provided the model for the velocity of an object in motion with variable mass Multi-Stage rocket allowed for higher altitudes(1914) Goddard applied three-axis control, gyroscopes, steerable thrust to rockets Used de Laval nozzles for increased performance (1914) Newton’s laws, especially his second and third, formed the basis of rocket science and helped rocket scientists advance their designs until today. William Hale created a far more accurate rocket by using vanes that directed the exhaust, creating a spin that made the rocket more precise.

Tsiolkovsky’s rocket Equation Newton’s 2nd Law cannot describe variable-mass systems Tsiolkovsky used conversation of momentum to model rocket velocities necessary for space travel 𝐹= 𝑑 𝑚𝑣 𝑑𝑡 =𝑚 𝑑𝑣 𝑑𝑡 +𝑣 𝑒 𝑑𝑚 𝑑𝑡 ∆𝑉=𝑣 𝑒 𝑙𝑛 𝑚 0 𝑚 1 Used today to calculate change in rocket mass and velocity in a simple equation

Forces Acting on A Rocket On Center of Gravity Thrust Weight of Rocket On Center of Aerodynamic Pressure Lift Drag

Thrust 𝐹 𝑇ℎ𝑟𝑢𝑠𝑡 = 𝑚 𝑣 𝑒 + ( 𝑝 𝑒 - 𝑝 0 ) * 𝐴 𝑒 Application of Newton’s Third Law Exhaust is forced down through nozzle, pushes rocket back up 𝐹 𝑇ℎ𝑟𝑢𝑠𝑡 = 𝑚 𝑣 𝑒 + ( 𝑝 𝑒 - 𝑝 0 ) * 𝐴 𝑒 Thrust is proportional to velocity of exhaust and rate of flow Can be maximized by increasing velocity or mass flow rate

Drag Types of Drag: Difference in Pressure Density of Air Turbulent Flow Frictional Drag Caused by friction of air with body of rocket Proportional to velocity Force of drag equation is one half of ro, the viscosity and density of the air, times the area of the body times velocity squared. Ro and velocity are dynamic variables, while the area and the drag coefficient are constant

Specific Impulse 𝐼 𝑠𝑝 = 𝐹 𝑇ℎ𝑟𝑢𝑠𝑡 𝑚𝑔 Measure of efficiency of rocket engines Force of thrust generated per unit propellant flow rate Higher specific impulse means very efficient rocket Can be used to measure thrust 𝐼 𝑠𝑝 = 𝐹 𝑇ℎ𝑟𝑢𝑠𝑡 𝑚𝑔

Rocket control 6 degrees of movement Pitch, yaw, roll Gimbaled thrust Nozzle directs force to rotate around Center of Mass Vernier Rocket Uses smaller rockets to orient

Other types of Rocket Engines Ion thrusters Very low thrust but extremely high specific impulse Use electric potential forces to accelerate ions to 50km/s Dawn spacecraft used 275 kg of propellant to accelerate to 10km/s Downsides: Takes a long time to accelerate Only functional in space Could be used to travel near light speed for interstellar missions Deep Space 1’s ion thruster

Works Cited Benson, Tom. "Brief History of Rockets." Brief History of Rockets. NASA, 12 June 2014. Web. 30 July 2015. Peraire, J. Variable Mass Systems – The Rocket Equation. MIT OpenCourseWare. 2008. Web. 30 July 2015. Rogers, Lucy. It's Only Rocket Science: An Introduction in Plain English. New York: Springer, 2008. Web. Tewari, Ashish. Advanced Control of Aircraft, Spacecraft, and Rockets. Chichester, West Sussex, U.K.: Wiley, 2011. Print.

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