A Teacher’s Guide to Helping Students Simulate the Flight of Model Rockets (Version 2)

1. A Teacher’s Guide to Helping Students Simulate the Flight of Model Rockets(Version 2) Kyle Voge Akins High School, Austin, Texas kvoge@austinisd.org 10-28-2007

2. Introduction • This program is intended for high school classes- the math is probably too difficult for middle school students. • There are software programs available to predict how high and fast a rocket will fly- but using them doesn’t allow students to understand the MATH behind rocketry.

3. Introduction • The purpose of this program is for students to understand the kinetic and kinematic details of a rocket in flight. • But first, a little physics….

4. Introduction • Most of the time a person builds and flies a rocket, they are concerned with only two things: Altitude and Velocity. • In other words, • How high did it fly? • How fast did it go?

5. Introduction • In order to calculate how high and how fast a rocket flies, one must determine the acceleration of the rocket. • To get acceleration, we need to know the net force. • To get net force, we must determine each individual force acting on the rocket.

6. Introduction • So, to calculate the velocity and altitude of a rocket, we: • Find each force acting on the rocket • Find the net force on the rocket • Determine the acceleration of the rocket Not too difficult, yet…

7. Introduction • Once we find the acceleration of the rocket, it is quite easy to find the velocity and altitude. • We’ll start with finding the velocity…

8. Acceleration • Students learn in IPC that acceleration is equal to the change in velocity divided by the change in time: • Actually, it is a bit more complicated- acceleration is actually defined as the time derivative of velocity- but more on that later…

9. Acceleration • Regardless of how it’s written, acceleration describes HOW FAST THE VELOCITY IS CHANGING. • It stands to reason that: • ifyou know how fast the velocity of your rocket changes and • how much time it has been doing so, • then you can find how much the velocity changed.

10. Acceleration • Non-calculus based: • The acceleration equation can be rearranged to yield the change in velocity as shown below: • The result is quite simple: Multiply your acceleration by the time, and you get the change in velocity.

11. Acceleration • Non-calculus based: • So, change in velocity is equal to acceleration multiplied by time. • If your rocket accelerates for 1.5 seconds at a rate of 1.62 meters per second squared, the change in velocity is 2.43 meters per second

12. Acceleration This rocket accelerated at a constant rate of 1.62 m/s/s. After 1.5 seconds, it was traveling at 1.62*1.5 = 2.43 meters per second.

13. Acceleration Basically, you multiply length by height- the same technique to find the area of a rectangle.

14. Acceleration • That’s the gist of it- • The change in velocity at any point in time is equal to the area under the acceleration curve UP TO THAT TIME. • (That’s half of calculus……)

15. Acceleration • But what if your acceleration is not constant?

16. Acceleration • Calculus based: • The acceleration equation can be rearranged to yield the change in velocity as shown below: • From here, integrate both sides to get:

17. Acceleration • In other words, NOTHING IS TOO DIFFERENT FROM BEFORE! • The change in velocity is still THE AREA UNDER THE ACCELERATION GRAPH. • The only difference is that the area is no longer a rectangle….

18. Acceleration

19. Acceleration • So, to find the change in velocity for a rocket undergoing a changing acceleration you have two options: • Do the calculus • Pretend the curve is a rectangle (or LOTS of little rectangles…..)

20. Acceleration • What if we took a curve and pretended it was a lot of very skinny rectangles placed next to each other… • Wouldn’t it look like this?

21. Acceleration

22. Acceleration • If we did that, we could find the area of each rectangle and add them up. • It would not be EXACTLY the same as the “correct” calculus method, but it would be pretty close… and if we made LOTS of REALLY skinny rectangles, it would get even closer.

23. Acceleration • So let’s do it… let’s say that our rectangles are 0.1 seconds wide. • The first velocity is equal to the acceleration at the beginning of the first interval multiplied by the width of the interval PLUS the velocity from before… • V(1) = a(0)*0.1+0

24. Acceleration • The second velocity is equal to the acceleration at the beginning of THAT interval multiplied by the length of the interval PLUS the velocity from before… • V(2) = a(1)*0.1+V(1) • And so on….

25. Acceleration • V(1) = a(0)*0.1*0 • V(2) = a(1)*0.1+V(1) • V(3) = a(2)*0.1+V(2) • V(4) = a(3)*0.1+V(3) • Etc etc etc etc • V(j) = a(j-1)*0.1+V(j-1)

26. Acceleration • Finding the altitude is equally easy- just find the area under the velocity curve- or take a shortcut. • You know that V(j) = a(j-1)*0.1+V(j-1) • Yf = .5at2+Vit+Yi • So, Y(j) = 0.5*a(j-1)*0.1^2+V(j-1)*0.1+Y(j-1)

27. Acceleration • That pretty much wraps it up- you have the equations you need to find velocity and altitude. • Of course, that’s assuming you know the acceleration as a function of time…. But we still have to do that.

28. Introduction to Levels • There is a lot of complexity involved in this. • If you throw everything at the students at once, they will be lost. • I broke it into 6 levels of varying complexity. • You could work your way through all of them, or choose one as appropriate to your class’s ability.

29. Levels • Remember, we know how to find altitude and velocity from acceleration- but we don’t know the acceleration yet. • You need to find the net force on the rocket to find acceleration, and to find net force we need to identify the individual forces that act on it.

30. Levels • There are three forces acting on the rocket: • Weight • Thrust • Drag • The higher levels will incorporate more details of the effects of these as the complexity increases.

31. Levels • Level One: • Constant Weight • Constant Thrust • No Drag • Level Two: • Changing Weight • Constant Thrust • No Drag

32. Levels • Level Three • Changing Weight • Changing Thrust • No Drag • Level Four • Changing Weight • Changing Thrust • Drag is present, but simplified

33. Levels • Because Aerodynamic Drag is such a complicated force, Levels 4-6 will each seek to refine one aspect of the drag force equation. • Level Four • Constant density • Constant drag coefficient • Changing velocity

34. Levels • Level 5 • Changing density • Constant drag coefficient • Changing velocity • Level 6 • Changing density • Changing drag coefficient • Changing velocity

35. Levels • The way I worked this with my students was to have them work through about one level a week. • I used my rocket as an example and had them follow along with me as I made a spreadsheet in Excel. • They were then able to plug in the values for their rockets and get their results.

36. Level One • In Excel, I set up columns for Time, Thrust, Mass, Weight, Net Force, Acceleration, Velocity, and Height • I didn’t really need to have columns for mass and weight- they don’t change in Level One, but it’ll make life easier later.

37. Level One • The mass of my rocket is 75 grams (0.075 kg). • The thrust of my Estes C6-5 motor is 6 Newtons for 1.8 seconds.

38. Level One • The time column is stepped by 0.1 seconds. • Thrust is 6 Newtons until 1.8 seconds, then 0 Newtons after that. • Mass is 0.075 kg the whole time • Weight is mass * gravity • Net Force is Thrust (up) minus Weight (down) • Acceleration is Net Force / mass (Newton’s Second Law)

39. Level One • The Velocity and Altitude equations from before get plugged in… of course, your first velocity and first altitude (at time = 0) are zero…. • The whole thing should look like this.

40. Level One

41. Level One • You only need to take the simulation out as far as max height- there’s no sense in calculating anything about the descent. • Just keep going until the velocity becomes negative- that means it’s turned around.

42. Level One • You can see that my rocket reached a max height of 927 meters (over 3000 feet!)

43. Level One

44. Level One, Summary • Students have a working (albeit inaccurate) model of rocket flight. • This is the level of math you would see on the AP Physics B exam- lots of simplifications, lots of assumptions. • We now get into math that is beyond anything they would otherwise see in high school.

45. Level Two • Level Two is identical to Level One is all aspects except for the changing mass. • As the motor fires, it expels a great deal of its mass- my Estes C6-5 motor loses 62% of its mass, and that is on the lower end of the spectrum- • The solid rocket booster (SRB) for the Space Shuttle is 88% fuel!

46. Level Two • So, the mass of our rocket is changing throughout flight. • That means the weight is changing, so the net force is changing, so we now have a new acceleration profile. • That doesn’t change HOW we get velocity and altitude… it just changes what they are.

47. Level Two • Students can determine the mass loss for themselves using a triple beam balance- but for now, I’ll just tell you that a full C6-5 motor has a mass of 24 grams, and that drops to 9 grams after the motor has burned out. • That means the whole rocket loses 15 grams of mass during flight- and all 15 grams are lost during the 1.8 seconds that the motor takes to fire.

48. Level Two • We need to find the mass of the rocket at any point in time- this is a great algebra review for the kids- it incorporates both piece-wise defined functions and the mathematics of rates. • We’ll start by calculating the mass flow rate- this is how fast the motor loses mass while it is burning.

49. Level Two • My motor loses 0.015 kg in 1.8 seconds. • That means it has a mass flow rate of -0.015/1.8 = -.00833 kg/s. • All we have to do now is write the equation that gives us the mass at any point during the first 1.8 seconds.

50. Level Two • You could get into a discussion at this point about lines- what “y = mx+b” really means and how that applies to this situation. • I’ll skip to the chase- If you plot mass vs. time for my rocket it looks something like this.