Transcript Document
Applied Math Series: Mathematics and Model Rocketry It Doesn't Take a Rocket Scientist to Do Rocket Science Brian Cheek Centerville High School Applied Math with Mr. Cheek Introduction Why are we doing this? To explore how math can be applied to things that happen in the real world To help you understand how to use technology to make the job easier To convince you that math can actually be useful – and maybe even fun! Applied Math with Mr. Cheek The Basic Process We’ll be going through a 10-step unit during your problem-solving time about model rocketry And the 10 steps are… Applied Math with Mr. Cheek The Basic Process Basic concepts: setting up a spreadsheet The three basic forces that act on a rocket Thrust and gravity Air resistance Newton’s 2nd Law of Motion Applied Math with Mr. Cheek The Basic Process Finding acceleration Finding velocity from acceleration Finding altitude from velocity Putting it all together Launching a model rocket! Applied Math with Mr. Cheek Basic Concepts: setting up a spreadsheet Enter all the constants – they describe facts about your rocket, motor, the air, etc. Set up column titles – they are the categories that you’ll teach the spread sheet to compute Set up the time increments – we’ll be computing what a rocket does every tenth of a second! Applied Math with Mr. Cheek Basic Concepts: setting up a spreadsheet Here’s an example Motor C6-5 Burn time 1.86 Air Density 1.22 Coast time 4.28 Rocket (loaded) Prop. Mass 0.0108 Rocket (unloaded) Casing mass 0.0133 Gravity -9.81 Rocket Mongoose Area Diameter 0.0248 Mass 0.0398 Drag coef. 0.75 Time Thrust Gravity Drag Total Force Mass Applied Math with Mr. Cheek Accel. Velocity Altitude Basic Concepts 1. Lift-off 2. Motor thrust – acceleration 3. Motor burnout – coast phase 4. Tracking smoke 5. Ejection charge at apogee 6. Recovery system deployed 7. Rocket slowly descends 8. Rocket recovery Applied Math with Mr. Cheek Mass The problem: mass changes during flight - fuel is burned over time, so the rocket gets lighter The solution: take the average. Find mass of pre-launch rocket Find mass of rocket when fuel is all burned out Add these together, divide by two Applied Math with Mr. Cheek Mass (continued) Use this average until motor burnout time After motor burnout, use the value for the mass of the rocket when fuel is all burned out Applied Math with Mr. Cheek The Three Forces that Act on a Rocket Motor Thrust Gravity (weight) Air Resistance (drag) Applied Math with Mr. Cheek The Three Forces that Act on a Rocket Motor Thrust Thrust = “how hard the rocket motor is pushing the rocket upward” Measured in Newtons One pound = 4.45 N We read average thrust right off the outside of the motor Applied Math with Mr. Cheek The Three Forces that Act on a Rocket Gravity (weight) Force with which gravity pulls you down depends on your mass However, gravity causes all falling objects to accelerate at the same rate This rate is called the Gravitational Constant = -9.81 Applied Math with Mr. Cheek The Three Forces that Act on a Rocket Air resistance (drag) Drag = “how hard the air is pushing against you, trying to slow you down” Depends on your size, shape, how fast you are going, and how dense the air is. Applied Math with Mr. Cheek Thrust Read from the motor casing In our example, “C6-5”, the 6 gives the average thrust of the motor This motor pushes the rocket upward with a force of 6 Newtons Enter a “6” in all the cells in the “Thrust” column until motor burnout Enter a “0” in the rest of the Thrust column Applied Math with Mr. Cheek Gravity The force with which gravity pulls on an object is given by the equation F = mg m = the mass of the rocket (kg) g = acceleration caused by gravity (-9.81 m/sec2) Applied Math with Mr. Cheek Air Resistance (Drag) We get to “steal” a formula figured out by experts in fluid mechanics The formula is Drag = 0.5CdAV2 = Air density = 1.2 kg/m3 A = Cross-sectional area of rocket V = velocity Applied Math with Mr. Cheek Total Net Force We have computed the forces due to: Thrust Gravity Drag The sum of those forces is the Total Net Force on the rocket Applied Math with Mr. Cheek Newton’s 2nd Law Newton’s second law states that sum of all forces acting on an object is equal to the product of its mass and its acceleration, i.e. F = ma Since we know F and m, we can easily compute a Applied Math with Mr. Cheek Computing Acceleration Acceleration describes how much an object’s velocity changes each second Simply take the computed force acting on the rocket and divide it by the computed mass of the rocket Applied Math with Mr. Cheek Computing Velocity Velocity describes how much our rocket’s altitude changes each second Current velocity = previous velocity plus current acceleration Since we are computing the change in altitude every tenth of a second, we need to multiply current acceleration by 0.1 Applied Math with Mr. Cheek Computing Altitude Current altitude = previous altitude plus current velocity Since we are computing the change in altitude every tenth of a second, we need to multiply current velocity by 0.1 Applied Math with Mr. Cheek You’ve done it! Now you can simply use the spreadsheet to look up the acceleration, velocity, and altitude Acceleration is in m/sec2 Velocity is in m/sec Altitude is in meters Want to convert these to more familiar units? Applied Math with Mr. Cheek Unit Conversion To change velocity from m/sec to mph, multiply by 2.237 To change meters to feet, multiply by 3.28 Applied Math with Mr. Cheek