a teacher’s guide to helping students simulate the flight of model rockets (version 2) kyle voge...

A Teacher’s Guide to Helping A Teacher’s Guide to Helping Students Simulate the Flight of Students Simulate the Flight of

Model RocketsModel Rockets(Version 2)(Version 2)

Kyle VogeKyle VogeAkins High School, Austin, TexasAkins High School, Austin, Texas

[email protected]@austinisd.org10-28-200710-28-2007

mailto:[email protected]

IntroductionIntroduction

This program is intended for high school This program is intended for high school classes- the math is probably too difficult for classes- the math is probably too difficult for middle school students.middle school students.

There are software programs available to There are software programs available to predict how high and fast a rocket will fly- predict how high and fast a rocket will fly- but using them doesn’t allow students to but using them doesn’t allow students to understand the MATH behind rocketry.understand the MATH behind rocketry.


The purpose of this program is for students The purpose of this program is for students to understand the kinetic and kinematic to understand the kinetic and kinematic details of a rocket in flight.details of a rocket in flight.

But first, a little physics….But first, a little physics….


Most of the time a person builds and flies a Most of the time a person builds and flies a rocket, they are concerned with only two rocket, they are concerned with only two things: Altitude and Velocity.things: Altitude and Velocity.

In other words, In other words, – How high did it fly?How high did it fly?– How fast did it go?How fast did it go?


In order to calculate how high and how fast In order to calculate how high and how fast a rocket flies, one must determine the a rocket flies, one must determine the acceleration of the rocket. acceleration of the rocket.

To get acceleration, we need to know the To get acceleration, we need to know the net force.net force.

To get net force, we must determine each To get net force, we must determine each individual force acting on the rocket.individual force acting on the rocket.


So, to calculate the velocity and altitude of So, to calculate the velocity and altitude of a rocket, we:a rocket, we:

1)1) Find each force acting on the rocketFind each force acting on the rocket

2)2) Find the net force on the rocketFind the net force on the rocket

3)3) Determine the acceleration of the rocketDetermine the acceleration of the rocket

Not too difficult, yet…Not too difficult, yet…


Once we find the acceleration of the rocket, Once we find the acceleration of the rocket, it is quite easy to find the velocity and it is quite easy to find the velocity and altitude. altitude.

We’ll start with finding the velocity…We’ll start with finding the velocity…

AccelerationAcceleration

Students learn in IPC that acceleration is Students learn in IPC that acceleration is equal to the change in velocity divided by the equal to the change in velocity divided by the change in time:change in time:

Actually, it is a bit more complicated- Actually, it is a bit more complicated- acceleration is actually defined as the time acceleration is actually defined as the time derivative of velocity- but more on that later…derivative of velocity- but more on that later…

t

Va

dt

dVa


Regardless of how it’s written, acceleration Regardless of how it’s written, acceleration describes HOW FAST THE VELOCITY IS describes HOW FAST THE VELOCITY IS CHANGING.CHANGING.

It stands to reason that:It stands to reason that: ifif you know how fast the velocity of your rocket you know how fast the velocity of your rocket

changes changes andand how much time it has been doing so, how much time it has been doing so, thenthen you can find how much the velocity changed. you can find how much the velocity changed.

AccelerationAcceleration Non-calculus based:Non-calculus based: The acceleration equation can be rearranged to The acceleration equation can be rearranged to

yield the change in velocity as shown below:yield the change in velocity as shown below:

The result is quite simple: Multiply your The result is quite simple: Multiply your acceleration by the time, and you get the change acceleration by the time, and you get the change in velocity.in velocity.

t

Va

taV


Non-calculus based:Non-calculus based: So, change in velocity is equal to acceleration So, change in velocity is equal to acceleration

multiplied by time.multiplied by time. If your rocket accelerates for 1.5 seconds at a If your rocket accelerates for 1.5 seconds at a

rate of 1.62 meters per second squared, the rate of 1.62 meters per second squared, the change in velocity is 2.43 meters per secondchange in velocity is 2.43 meters per second

AccelerationAccelerationAcceleration vs. Time

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3

Time (seconds)

Acc

eler

atio

n (

m/s

/s)

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.


0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3

Time (seconds)

Acc

eler

atio

n (

m/s

/s)

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


That’s the gist of it- That’s the gist of it- The change in velocity at any point in The change in velocity at any point in

time is equal to the area under the time is equal to the area under the acceleration curve UP TO THAT TIME.acceleration curve UP TO THAT TIME.

(That’s half of calculus……)(That’s half of calculus……)


But what if your acceleration is not But what if your acceleration is not constant?constant?

AccelerationAcceleration Calculus based:Calculus based: The acceleration equation can be rearranged to The acceleration equation can be rearranged to

yield the change in velocity as shown below:yield the change in velocity as shown below:

From here, integrate both sides to get:From here, integrate both sides to get:dt

dVa dtadV

dtadV dtaV


In other words, NOTHING IS TOO In other words, NOTHING IS TOO DIFFERENT FROM BEFORE!DIFFERENT FROM BEFORE!

The change in velocity is still THE AREA The change in velocity is still THE AREA UNDER THE ACCELERATION GRAPH.UNDER THE ACCELERATION GRAPH.

The only difference is that the area is no The only difference is that the area is no longer a rectangle…. longer a rectangle….


Acceleration vs. Time

0

0.5

1

1.5

2

2.5

Time (seconds)

Ac

ce

lera

tio

n (

m/s

/s)


So, to find the change in velocity for a So, to find the change in velocity for a rocket undergoing a changing rocket undergoing a changing acceleration you have two options:acceleration you have two options: Do the calculusDo the calculus Pretend the curve is a rectangle (or LOTS of Pretend the curve is a rectangle (or LOTS of

little rectangles…..)little rectangles…..)


What if we took a curve and pretended it What if we took a curve and pretended it was a lot of very skinny rectangles was a lot of very skinny rectangles placed next to each other…placed next to each other…

Wouldn’t it look like this?Wouldn’t it look like this?


0

0.5

1

1.5

2

2.5

Time (seconds)

Ac

ce

lera

tio

n (

m/s

/s)


If we did that, we could find the area of If we did that, we could find the area of each rectangle and add them up. each rectangle and add them up.

It would not be EXACTLY the same as It would not be EXACTLY the same as the “correct” calculus method, but it the “correct” calculus method, but it would be pretty close… and if we made would be pretty close… and if we made LOTS of REALLY skinny rectangles, it LOTS of REALLY skinny rectangles, it would get even closer. would get even closer.


So let’s do it… let’s say that our So let’s do it… let’s say that our rectangles are 0.1 seconds wide. rectangles are 0.1 seconds wide.

The first velocity is equal to the The first velocity is equal to the acceleration at the beginning of the first acceleration at the beginning of the first interval multiplied by the width of the interval multiplied by the width of the interval PLUS the velocity from before… interval PLUS the velocity from before…

V(1) = a(0)*0.1+0V(1) = a(0)*0.1+0


The second velocity is equal to the The second velocity is equal to the acceleration at the beginning of THAT acceleration at the beginning of THAT interval multiplied by the length of the interval multiplied by the length of the interval PLUS the velocity from before…interval PLUS the velocity from before…

V(2) = a(1)*0.1+V(1)V(2) = a(1)*0.1+V(1) And so on….And so on….


V(1) = a(0)*0.1*0V(1) = a(0)*0.1*0 V(2) = a(1)*0.1+V(1)V(2) = a(1)*0.1+V(1) V(3) = a(2)*0.1+V(2)V(3) = a(2)*0.1+V(2) V(4) = a(3)*0.1+V(3)V(4) = a(3)*0.1+V(3) Etc etc etc etcEtc etc etc etc

V(j) = a(j-1)*0.1+V(j-1)V(j) = a(j-1)*0.1+V(j-1)

AccelerationAcceleration Finding the altitude is equally easy- just find Finding the altitude is equally easy- just find

the area under the velocity curve- or take a the area under the velocity curve- or take a shortcut. shortcut.

You know that V(j) = a(j-1)*0.1+V(j-1)You know that V(j) = a(j-1)*0.1+V(j-1)

YYff = .5at = .5at22+V+Viit+Yt+Yii

So, Y(j) = 0.5*a(j-1)*0.1^2+V(j-1)*0.1+Y(j-1)So, Y(j) = 0.5*a(j-1)*0.1^2+V(j-1)*0.1+Y(j-1)


That pretty much wraps it up- you have That pretty much wraps it up- you have the equations you need to find velocity the equations you need to find velocity and altitude. and altitude.

Of course, that’s assuming you know the Of course, that’s assuming you know the acceleration as a function of time…. But acceleration as a function of time…. But we still have to do that.we still have to do that.

Introduction to LevelsIntroduction to LevelsThere is a lot of complexity involved in There is a lot of complexity involved in this.this.

If you throw everything at the students at If you throw everything at the students at once, they will be lost.once, they will be lost.

I broke it into 6 levels of varying I broke it into 6 levels of varying complexity.complexity.

You could work your way through all of You could work your way through all of them, or choose one as appropriate to them, or choose one as appropriate to your class’s ability.your class’s ability.

LevelsLevels

Remember, we know how to find Remember, we know how to find altitude and velocity from altitude and velocity from acceleration- but we don’t know the acceleration- but we don’t know the acceleration yet.acceleration yet.

You need to find the net force on the You need to find the net force on the rocket to find acceleration, and to rocket to find acceleration, and to find net force we need to identify the find net force we need to identify the individual forces that act on it.individual forces that act on it.

LevelsLevelsThere are three forces acting on the There are three forces acting on the rocket:rocket:– WeightWeight– ThrustThrust– DragDrag

The higher levels will incorporate The higher levels will incorporate more details of the effects of these more details of the effects of these as the complexity increases.as the complexity increases.

LevelsLevelsLevel One:Level One:– Constant WeightConstant Weight– Constant ThrustConstant Thrust– No DragNo Drag

Level Two:Level Two:– Changing WeightChanging Weight– Constant ThrustConstant Thrust– No DragNo Drag

LevelsLevelsLevel ThreeLevel Three– Changing WeightChanging Weight– Changing ThrustChanging Thrust– No DragNo Drag

Level FourLevel Four– Changing WeightChanging Weight– Changing ThrustChanging Thrust– Drag is present, but simplifiedDrag is present, but simplified

LevelsLevels

Because Aerodynamic Drag is such a Because Aerodynamic Drag is such a complicated force, Levels 4-6 will complicated force, Levels 4-6 will each seek to refine one aspect of the each seek to refine one aspect of the drag force equation.drag force equation.

Level FourLevel Four– Constant densityConstant density– Constant drag coefficientConstant drag coefficient– Changing velocityChanging velocity

LevelsLevels

Level 5Level 5– Changing densityChanging density– Constant drag coefficientConstant drag coefficient– Changing velocityChanging velocity

Level 6Level 6– Changing densityChanging density– Changing drag coefficientChanging drag coefficient– Changing velocityChanging velocity

LevelsLevelsThe way I worked this with my students The way I worked this with my students was to have them work through about one was to have them work through about one level a week. level a week.

I used my rocket as an example and had I used my rocket as an example and had them follow along with me as I made a them follow along with me as I made a spreadsheet in Excel.spreadsheet in Excel.

They were then able to plug in the values They were then able to plug in the values for their rockets and get their results.for their rockets and get their results.

Level OneLevel One

In Excel, I set up columns for Time, In Excel, I set up columns for Time, Thrust, Mass, Weight, Net Force, Thrust, Mass, Weight, Net Force, Acceleration, Velocity, and HeightAcceleration, Velocity, and Height

I didn’t really need to have columns I didn’t really need to have columns for mass and weight- they don’t for mass and weight- they don’t change in Level One, but it’ll make change in Level One, but it’ll make life easier later.life easier later.

Level OneLevel One

The mass of my rocket is 75 grams The mass of my rocket is 75 grams (0.075 kg).(0.075 kg).

The thrust of my Estes C6-5 motor is The thrust of my Estes C6-5 motor is 6 Newtons for 1.8 seconds.6 Newtons for 1.8 seconds.

Level OneLevel One

The time column is stepped by 0.1 seconds.The time column is stepped by 0.1 seconds. Thrust is 6 Newtons until 1.8 seconds, then Thrust is 6 Newtons until 1.8 seconds, then

0 Newtons after that.0 Newtons after that. Mass is 0.075 kg the whole timeMass is 0.075 kg the whole time Weight is mass * gravityWeight is mass * gravity Net Force is Thrust (up) minus Weight Net Force is Thrust (up) minus Weight

(down)(down) Acceleration is Net Force / mass (Newton’s Acceleration is Net Force / mass (Newton’s

Second Law)Second Law)

Level OneLevel One

The Velocity and Altitude equations The Velocity and Altitude equations from before get plugged in… of from before get plugged in… of course, your first velocity and first course, your first velocity and first altitude (at time = 0) are zero…. altitude (at time = 0) are zero….

The whole thing should look like this.The whole thing should look like this.

Level OneLevel OneTime Thrust Mass Weight Net Force Acceleration Velocity Height

0 6 0.075 0.73575 5.26425 70.19 0 00.1 6 0.075 0.73575 5.26425 70.19 7.019 0.350950.2 6 0.075 0.73575 5.26425 70.19 14.038 1.40380.3 6 0.075 0.73575 5.26425 70.19 21.057 3.158550.4 6 0.075 0.73575 5.26425 70.19 28.076 5.61520.5 6 0.075 0.73575 5.26425 70.19 35.095 8.773750.6 6 0.075 0.73575 5.26425 70.19 42.114 12.63420.7 6 0.075 0.73575 5.26425 70.19 49.133 17.196550.8 6 0.075 0.73575 5.26425 70.19 56.152 22.46080.9 6 0.075 0.73575 5.26425 70.19 63.171 28.42695

1 6 0.075 0.73575 5.26425 70.19 70.19 35.0951.1 6 0.075 0.73575 5.26425 70.19 77.209 42.464951.2 6 0.075 0.73575 5.26425 70.19 84.228 50.53681.3 6 0.075 0.73575 5.26425 70.19 91.247 59.310551.4 6 0.075 0.73575 5.26425 70.19 98.266 68.78621.5 6 0.075 0.73575 5.26425 70.19 105.285 78.963751.6 6 0.075 0.73575 5.26425 70.19 112.304 89.84321.7 6 0.075 0.73575 5.26425 70.19 119.323 101.424551.8 0 0.075 0.73575 -0.73575 -9.81 126.342 113.70781.9 0 0.075 0.73575 -0.73575 -9.81 125.361 126.29295

2 0 0.075 0.73575 -0.73575 -9.81 124.38 138.782.1 0 0.075 0.73575 -0.73575 -9.81 123.399 151.16895

Level OneLevel One

You only need to take the simulation out You only need to take the simulation out as far as max height- there’s no sense in as far as max height- there’s no sense in calculating anything about the descent.calculating anything about the descent.

Just keep going until the velocity Just keep going until the velocity becomes negative- that means it’s becomes negative- that means it’s turned around.turned around.

Level OneLevel One

You can see that my rocket reached a You can see that my rocket reached a max height of 927 meters (over 3000 max height of 927 meters (over 3000 feet!)feet!)

14.2 0 0.075 0.73575 -0.73575 -9.81 4.698 926.155814.3 0 0.075 0.73575 -0.73575 -9.81 3.717 926.5765514.4 0 0.075 0.73575 -0.73575 -9.81 2.736 926.899214.5 0 0.075 0.73575 -0.73575 -9.81 1.755 927.1237514.6 0 0.075 0.73575 -0.73575 -9.81 0.774 927.250214.7 0 0.075 0.73575 -0.73575 -9.81 -0.207 927.2785514.8 0 0.075 0.73575 -0.73575 -9.81 -1.188 927.2088

Time Thrust Mass Weight Net Force Acceleration Velocity Height

Level OneLevel One

Leve l One Altitude and Ve locity vs. Time

-200

0

200

400

600

800

1000

0 2 4 6 8 10 12 14 16

Time (seconds)

Me

ters

or

m/s

Velocity Height

Level One, SummaryLevel One, Summary

Students have a working (albeit inaccurate) Students have a working (albeit inaccurate) model of rocket flight. model of rocket flight.

This is the level of math you would see on This is the level of math you would see on the AP Physics B exam- lots of the AP Physics B exam- lots of simplifications, lots of assumptions.simplifications, lots of assumptions.

We now get into math that is beyond We now get into math that is beyond anything they would otherwise see in high anything they would otherwise see in high school.school.

Level TwoLevel Two

Level Two is identical to Level One is all aspects Level Two is identical to Level One is all aspects except for the changing mass. except for the changing mass.

As the motor fires, it expels a great deal of its As the motor fires, it expels a great deal of its mass- my Estes C6-5 motor loses 62% of its mass- my Estes C6-5 motor loses 62% of its mass, and that is on the lower end of the mass, and that is on the lower end of the spectrum-spectrum-

The solid rocket booster (SRB) for the Space The solid rocket booster (SRB) for the Space Shuttle is 88% fuel!Shuttle is 88% fuel!

Level TwoLevel Two

So, the mass of our rocket is changing So, the mass of our rocket is changing throughout flight. throughout flight.

That means the weight is changing, so the That means the weight is changing, so the net force is changing, so we now have a net force is changing, so we now have a new acceleration profile. new acceleration profile.

That doesn’t change HOW we get velocity That doesn’t change HOW we get velocity and altitude… it just changes what they are.and altitude… it just changes what they are.

Level TwoLevel Two

Students can determine the mass loss for Students can determine the mass loss for themselves using a triple beam balance- but for themselves using a triple beam balance- but for now, I’ll just tell you that a full C6-5 motor has a 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 mass of 24 grams, and that drops to 9 grams after the motor has burned out.the motor has burned out.

That means the whole rocket loses 15 grams of That means the whole rocket loses 15 grams of mass during flight- and all 15 grams are lost during mass during flight- and all 15 grams are lost during the 1.8 seconds that the motor takes to fire.the 1.8 seconds that the motor takes to fire.

Level TwoLevel Two

We need to find the mass of the rocket at We need to find the mass of the rocket at any point in time- this is a great algebra any point in time- this is a great algebra review for the kids- it incorporates both review for the kids- it incorporates both piece-wise defined functions and the piece-wise defined functions and the mathematics of rates.mathematics of rates.

We’ll start by calculating the mass flow rate- We’ll start by calculating the mass flow rate- this is how fast the motor loses mass while it this is how fast the motor loses mass while it is burning.is burning.

Level TwoLevel Two

My motor loses 0.015 kg in 1.8 seconds.My motor loses 0.015 kg in 1.8 seconds.

That means it has a mass flow rate of That means it has a mass flow rate of -0.015/1.8 = -.00833 kg/s.-0.015/1.8 = -.00833 kg/s.

All we have to do now is write the equation All we have to do now is write the equation that gives us the mass at any point during that gives us the mass at any point during the first 1.8 seconds.the first 1.8 seconds.

Level TwoLevel Two

You could get into a discussion at this point You could get into a discussion at this point about lines- what “y = mx+b” really means about lines- what “y = mx+b” really means and how that applies to this situation. and how that applies to this situation.

I’ll skip to the chase- If you plot mass vs. I’ll skip to the chase- If you plot mass vs. time for my rocket it looks something like time for my rocket it looks something like this.this.

Level TwoLevel Two

Mass vs. Time

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.5 1 1.5 2 2.5 3 3.5

Time (s)

Mas

s (k

g)

You can see this is a piece-wise defined You can see this is a piece-wise defined function. While the motor burns, it loses function. While the motor burns, it loses mass. After it burns out, mass is constant.mass. After it burns out, mass is constant.

Level TwoLevel Two

While the motor is burning, the mass can be While the motor is burning, the mass can be found using a linear equation:found using a linear equation:Y = mx+b… but our “Y” is mass, “m” is the mass Y = mx+b… but our “Y” is mass, “m” is the mass

flow rate, “x” is time, and “b” is the initial mass.flow rate, “x” is time, and “b” is the initial mass.

So, we have: mass = mdot*time+initial massSo, we have: mass = mdot*time+initial mass (mdot = mass flow rate)(mdot = mass flow rate)

Level TwoLevel Two

In other words, In other words,

Mass = (-0.008333)*time+0.075Mass = (-0.008333)*time+0.075

Of course, this is ONLY for the first 1.8 seconds. After that, Of course, this is ONLY for the first 1.8 seconds. After that, the mass of the rocket is just the initial mass – 0.015.the mass of the rocket is just the initial mass – 0.015.

For this example, the “final” mass is 0.06kg.For this example, the “final” mass is 0.06kg.

And, of course, the weight at any point in time is equal to And, of course, the weight at any point in time is equal to the mass at that point in time multiplied by gravity.the mass at that point in time multiplied by gravity.

Level TwoLevel Two

Everything else in the Excel spreadsheet Everything else in the Excel spreadsheet should be the same- in fact, I have students should be the same- in fact, I have students just save LevelOne.xls as LevelTwo.xls and just save LevelOne.xls as LevelTwo.xls and change the mass formula.change the mass formula.

I put the spreadsheet on the next slide so I put the spreadsheet on the next slide so you can see how the mass changes. you can see how the mass changes.

Level TwoLevel TwoTime Thrust Mass Weight Net Force Acceleration Velocity Height

0 6 0.075 0.73575 5.26425 70.19 0 00.1 6 0.074 0.72758 5.27242 71.088876 7.019 0.350950.2 6 0.073 0.7194 5.2806 72.008182 14.1279 1.40829440.3 6 0.073 0.71123 5.28877 72.948621 21.3287 3.18112410.4 6 0.072 0.70305 5.29695 73.91093 28.6236 5.67873770.5 6 0.071 0.69488 5.30512 74.895882 36.0147 8.91064920.6 6 0.07 0.6867 5.3133 75.904286 43.5042 12.8865950.7 6 0.069 0.67853 5.32147 76.936988 51.0947 17.6165410.8 6 0.068 0.67035 5.32965 77.994878 58.7884 23.1106940.9 6 0.068 0.66218 5.33782 79.078889 66.5879 29.379506

1 6 0.067 0.654 5.346 80.19 74.4958 36.4336871.1 6 0.066 0.64583 5.35417 81.329241 82.5148 44.2842121.2 6 0.065 0.63765 5.36235 82.497692 90.6477 52.9423341.3 6 0.064 0.62948 5.37052 83.696494 98.8974 62.419591.4 6 0.063 0.6213 5.3787 84.926842 107.267 72.7278171.5 6 0.063 0.61313 5.38687 86.19 115.76 83.8791611.6 6 0.062 0.60495 5.39505 87.487297 124.379 95.8860891.7 6 0.061 0.59678 5.40322 88.820137 133.128 108.76141.8 0 0.06 0.5886 -0.5886 -9.81 142.01 122.518251.9 0 0.06 0.5886 -0.5886 -9.81 141.029 136.67016

2 0 0.06 0.5886 -0.5886 -9.81 140.048 150.72396

Level TwoLevel Two

Again, just take the simulation out until just after Again, just take the simulation out until just after max height- we don’t care about the descent of max height- we don’t care about the descent of the rocket.the rocket.

My rocket went much higher this time- over 200 My rocket went much higher this time- over 200 meters higher!meters higher!

15.9 0 0.06 0.5886 -0.5886 -9.81 3.68852 1149.689516 0 0.06 0.5886 -0.5886 -9.81 2.70752 1150.0093

16.1 0 0.06 0.5886 -0.5886 -9.81 1.72652 1150.23116.2 0 0.06 0.5886 -0.5886 -9.81 0.74552 1150.354616.3 0 0.06 0.5886 -0.5886 -9.81 -0.2355 1150.380116.4 0 0.06 0.5886 -0.5886 -9.81 -1.2165 1150.3075


Level TwoLevel TwoLevel Two Altitude and Velocity vs. Time

-200

0

200

400

600

800

1000

1200

1400

0 5 10 15 20

Time (seconds)

Met

ers

or

m/s

Velocity

Height

Level TwoLevel TwoComparison of the First Two Levels of Simulations

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12 14 16 18

Time (seconds)

Alt

itu

de

(m

ete

rs)

Level One Level Two

Level TwoLevel Two

It makes sense that this rocket went higher.It makes sense that this rocket went higher.

Newton’s 2Newton’s 2ndnd law says F = m*a, so if the law says F = m*a, so if the force stays the same and the mass force stays the same and the mass decreases, the acceleration must increase.decreases, the acceleration must increase.

Level Two, SummaryLevel Two, Summary

This is an improvement over the Level One.This is an improvement over the Level One. We have an accurate model of one of the We have an accurate model of one of the

three forces (weight) acting on the rocket. three forces (weight) acting on the rocket. We still have a long way to go- the other two We still have a long way to go- the other two

forces (thrust and drag) are still “ideal” in our forces (thrust and drag) are still “ideal” in our simulation- we need to make them more simulation- we need to make them more realistic.realistic.

Level ThreeLevel Three

Level Three is similar to Level Two in all Level Three is similar to Level Two in all aspects except for the changing thrust. aspects except for the changing thrust.

Model rocket motors (and high power Model rocket motors (and high power motors, for that matter) do NOT output a motors, for that matter) do NOT output a constant thrust- their thrust varies with constant thrust- their thrust varies with time.time.

In this level of simulation, we will include In this level of simulation, we will include this effect in our calculations.this effect in our calculations.


The only thing that will change in our The only thing that will change in our spreadsheet is the “Thrust” column. spreadsheet is the “Thrust” column.

Instead of being exactly 6 Newtons for Instead of being exactly 6 Newtons for the first 1.8 seconds, we will use the the first 1.8 seconds, we will use the actual value that the motor exerts.actual value that the motor exerts.

So the question becomes, So the question becomes, ““What is that value?”What is that value?”


Most high power motors have their thrust Most high power motors have their thrust data readily available on the internet. data readily available on the internet. Just go to Just go to www.thrustcurve.orgwww.thrustcurve.org and and search for the model you need.search for the model you need.

We, however, are interested in a very We, however, are interested in a very specific motor- an Estes C6-5- a motor specific motor- an Estes C6-5- a motor that this website does NOT have.that this website does NOT have.

It’s time to voyage into your school’s It’s time to voyage into your school’s physics lab and workshop areas…physics lab and workshop areas…

http://www.thrustcurve.org/


Most physics departments have either Most physics departments have either Vernier’s LoggerPro or Texas Intruments’ Vernier’s LoggerPro or Texas Intruments’ CBL data collection devices (or CBL data collection devices (or something similar).something similar).

Students can use the digital force probe Students can use the digital force probe to measure the thrust of an engine as it to measure the thrust of an engine as it burns.burns.


The only difficult part is to design a The only difficult part is to design a device that will hold the motor in place device that will hold the motor in place while it is firing- I put my students to the while it is firing- I put my students to the task of designing it. task of designing it.

It doesn’t have to be complicated- one It doesn’t have to be complicated- one group made it out of scrap wood and group made it out of scrap wood and PVC pipe.PVC pipe.


Use Google.com to find sample pictures Use Google.com to find sample pictures and schematics if you have trouble and schematics if you have trouble coming up with an idea.coming up with an idea.

However it gets done, eventually you will However it gets done, eventually you will have the thrust data- and most likely it have the thrust data- and most likely it will not be sampled every 1/10will not be sampled every 1/10thth of a of a second like our simulation was.second like our simulation was.


That leaves you with two choices- That leaves you with two choices- 1)1) Downsample the thrust data (take only the Downsample the thrust data (take only the

data points that correspond with 0.1 data points that correspond with 0.1 seconds, 0.2 seconds, 0.3 seconds, etc)seconds, 0.2 seconds, 0.3 seconds, etc)

2)2) Upsample the simulation (step your Upsample the simulation (step your spreadsheet by the same increment that the spreadsheet by the same increment that the force collection device used)force collection device used)

The thrust profile for my motor is shown on The thrust profile for my motor is shown on the next slide.the next slide.

Level ThreeLevel ThreeThrust Profile

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (seconds)

Th

rust

(N

ewto

ns)


The only thing that changes from Level The only thing that changes from Level Two is the “Thrust” column.Two is the “Thrust” column.

Have students save LevelTwo.xls as Have students save LevelTwo.xls as LevelThree.xls and replace all the 6 LevelThree.xls and replace all the 6 Newton values with the data obtained Newton values with the data obtained from the motor test.from the motor test.


Of course, every thrust value after 1.8 Of course, every thrust value after 1.8 seconds should still be 0 Newtons.seconds should still be 0 Newtons.

My spreadsheet for Level Three is shown My spreadsheet for Level Three is shown on the next slide.on the next slide.

Note that to save space and make the Note that to save space and make the table visible on one slide, I downsampled table visible on one slide, I downsampled the thrust data to 10 Hz.the thrust data to 10 Hz.

Level ThreeLevel ThreeTime Thrust Mass Weight Net Force Acceleration Velocity Height

0 0.946 0.075 0.73575 0.21025 2.8033333 0 00.1 4.826 0.074 0.72758 4.09842 55.259663 0.28033 0.01401670.2 9.936 0.073 0.7194 9.2166 125.68091 5.8063 0.31834830.3 14.09 0.073 0.71123 13.3788 184.53483 18.3744 1.52738280.4 11.446 0.072 0.70305 10.7429 149.90163 36.8279 4.2874960.5 7.381 0.071 0.69488 6.68612 94.392353 51.818 8.71979150.6 6.151 0.07 0.6867 5.4643 78.061429 61.2573 14.3735570.7 5.489 0.069 0.67853 4.81047 69.549036 69.0634 20.8895910.8 4.921 0.068 0.67035 4.25065 62.204634 76.0183 28.1436780.9 4.448 0.068 0.66218 3.78582 56.086296 82.2388 36.056533

1 4.258 0.067 0.654 3.604 54.06 87.8474 44.5608421.1 4.542 0.066 0.64583 3.89617 59.182405 93.2534 53.6158831.2 4.164 0.065 0.63765 3.52635 54.251538 99.1717 63.2371371.3 4.448 0.064 0.62948 3.81852 59.509481 104.597 73.4255591.4 4.448 0.063 0.6213 3.8267 60.421579 110.548 84.1827871.5 4.448 0.063 0.61313 3.83487 61.358 116.59 95.539671.6 2.933 0.062 0.60495 2.32805 37.752162 122.726 107.505451.7 1.325 0.061 0.59678 0.72822 11.970822 126.501 119.966781.8 0 0.06 0.5886 -0.5886 -9.81 127.698 132.676731.9 0 0.06 0.5886 -0.5886 -9.81 126.717 145.39748

2 0 0.06 0.5886 -0.5886 -9.81 125.736 158.020132.1 0 0.06 0.5886 -0.5886 -9.81 124.755 170.544682.2 0 0.06 0.5886 -0.5886 -9.81 123.774 182.97113


14.4 0 0.06 0.5886 -0.5886 -9.81 4.09201 962.9538514.5 0 0.06 0.5886 -0.5886 -9.81 3.11101 963.31414.6 0 0.06 0.5886 -0.5886 -9.81 2.13001 963.5760514.7 0 0.06 0.5886 -0.5886 -9.81 1.14901 963.7414.8 0 0.06 0.5886 -0.5886 -9.81 0.16801 963.8058614.9 0 0.06 0.5886 -0.5886 -9.81 -0.813 963.77361


You’ll see that this rocket didn’t go as high.You’ll see that this rocket didn’t go as high. That happened for several reasons, That happened for several reasons,

perhaps the most important being that the perhaps the most important being that the average thrust wasn’t quite 6 Newtons- it average thrust wasn’t quite 6 Newtons- it was closer to 5.3 Newtons.was closer to 5.3 Newtons.

Level ThreeLevel ThreeComparison of the First Three Levels of Simulations

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12 14 16 18

Time (seconds)

Alt

itu

de

(m

ete

rs)

Level One Level Two Level Three

Level Three, SummaryLevel Three, Summary

Level Three is the best yet- we now have Level Three is the best yet- we now have two of the three forces acting on the two of the three forces acting on the rocket modeled accurately.rocket modeled accurately.

All that is left now is to model the drag All that is left now is to model the drag force.force.

This takes us to Level Four…This takes us to Level Four…

Level FourLevel FourBecause the drag force is so complicated, Because the drag force is so complicated, it will take three Levels to fully model.it will take three Levels to fully model.

We’ll start by looking at the equation for We’ll start by looking at the equation for aerodynamic drag:aerodynamic drag:

2

2

1VACF DD

Level FourLevel FourA = effective cross A = effective cross sectional area (msectional area (m22))

CCDD = coefficient of drag = coefficient of drag (a number between 0 (a number between 0 and 1 that indicates how and 1 that indicates how sleek or streamlined the sleek or streamlined the rocket is)rocket is)

ΡΡ = density of air = density of air (kg/m(kg/m33))

V = velocityV = velocity

2

2

1VACF DD

Level FourLevel Four

A = effective cross sectional A = effective cross sectional area (marea (m22))

CCDD = coefficient of drag (a = coefficient of drag (a number between 0 and 1 number between 0 and 1 that indicates how sleek or that indicates how sleek or streamlined the rocket is)streamlined the rocket is)

ΡΡ = density of air (kg/m = density of air (kg/m33))

V = velocityV = velocity

This stays constantThis stays constant

Level SixLevel Six

Level FiveLevel Five


Here’s what we will change in each Here’s what we will change in each of the next three Levels:of the next three Levels:


The Area in this formula is just the The Area in this formula is just the cross-sectional area of the body cross-sectional area of the body tube- I used a 1” diameter tube tube- I used a 1” diameter tube (.0254 meters) so I used A = (.0254 meters) so I used A = pi*radius^2 to find the area of my pi*radius^2 to find the area of my rocket.rocket.

A = .000505 mA = .000505 m22


For now, we’ll assume air density to For now, we’ll assume air density to be a constant value (we’ll change it be a constant value (we’ll change it in Level Five)in Level Five)

P = 1.225 kg/mP = 1.225 kg/m33


We’ll also assume that the coefficient We’ll also assume that the coefficient of drag is a constant (we’ll determine of drag is a constant (we’ll determine its true value in Level Six- for now I’ll its true value in Level Six- for now I’ll just give it an arbitrary value)just give it an arbitrary value)

CCDD = 0.56 = 0.56


To calculate the effects of drag, we To calculate the effects of drag, we need to include it in the spreadsheet- need to include it in the spreadsheet- I made a new column called “Drag” I made a new column called “Drag” and included it in my “Net Force” and included it in my “Net Force” calculations.calculations.

Net Force = Thrust - Weight - DragNet Force = Thrust - Weight - Drag


I also added a column for air density.I also added a column for air density.

We will set that as a constant for We will set that as a constant for now, but in Level Five we will replace now, but in Level Five we will replace those constant values with “real” those constant values with “real” values.values.


Time Thrust Mass Weight Density Drag Net Force Acceleration Velocity Height0 0.946 0.075 0.73575 1.225 0 0.21025 2.8033333 0 0

0.1 4.826 0.074 0.72758 1.225 1.2E-05 4.09841 55.259499 0.28033 0.01401670.2 9.936 0.073 0.7194 1.225 0.00522 9.21138 125.60968 5.80628 0.31834750.3 14.09 0.073 0.71123 1.225 0.05227 13.3265 183.81385 18.3673 1.52702420.4 11.446 0.072 0.70305 1.225 0.20925 10.5337 146.98192 36.7486 4.28281850.5 7.381 0.071 0.69488 1.225 0.4101 6.27602 88.602693 51.4468 8.69259170.6 6.151 0.07 0.6867 1.225 0.56352 4.90078 70.011123 60.3071 14.2802880.7 5.489 0.069 0.67853 1.225 0.70196 4.10852 59.400281 67.3082 20.6610530.8 4.921 0.068 0.67035 1.225 0.83132 3.41933 50.038985 73.2482 27.6888760.9 4.448 0.068 0.66218 1.225 0.94878 2.83704 42.030283 78.2521 35.263894

1 4.258 0.067 0.654 1.225 1.05344 2.55056 38.25842 82.4552 43.2992591.1 4.542 0.066 0.64583 1.225 1.15346 2.74271 41.661437 86.281 51.736068

Level FourLevel FourYou see a drastic drop in performance here from You see a drastic drop in performance here from Level Three. (From 963 meters to 378 meters)Level Three. (From 963 meters to 378 meters)This is to be expected…. Drag plays a huge role. This is to be expected…. Drag plays a huge role. Think about it… how far can you throw a Think about it… how far can you throw a crumpled up piece of paper?crumpled up piece of paper?

7.8 0 0.06 0.5886 1.225 0.00302 -0.59162 -9.8603553 4.41582 377.560627.9 0 0.06 0.5886 1.225 0.00182 -0.59042 -9.8403778 3.42979 377.9529

8 0 0.06 0.5886 1.225 0.00093 -0.58953 -9.8254471 2.44575 378.246688.1 0 0.06 0.5886 1.225 0.00033 -0.58893 -9.8155288 1.4632 378.442128.2 0 0.06 0.5886 1.225 3.6E-05 -0.58864 -9.8105991 0.48165 378.539378.3 0 0.06 0.5886 1.225 3.9E-05 -0.58864 -9.8106441 -0.4994 378.53848

Time Thrust Mass Weight Density Drag Net Force Acceleration Velocity Height

Comparison of the First Four Levels of Simulations

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12 14 16 18

Time (seconds)

Alt

itu

de

(m

ete

rs)

Level One Level Two Level Three Level Five

Level FourLevel FourComparison of the First Four Levels of Simulations

0

200

400

600

800

1000

1200

1400

0 2 4 6 8 10 12 14 16 18

Time (seconds)

Alt

itu

de

(m

ete

rs)

Level One Level Two Level Three Level Four

Level Four, SummaryLevel Four, Summary

We’ve finally arrived at a simulation We’ve finally arrived at a simulation level that is “accurate”.level that is “accurate”.

Any changes from here on out will be Any changes from here on out will be much less drastic than those we’ve much less drastic than those we’ve seen before- more like minor seen before- more like minor tweaking than full scale changes.tweaking than full scale changes.


In Level Five, we will consider the change in air In Level Five, we will consider the change in air density with altitude. density with altitude.

You know that in higher elevations the air is You know that in higher elevations the air is thinner- that plays a role in the drag calculations.thinner- that plays a role in the drag calculations.

As the rocket flies higher, the air gets thinner, so As the rocket flies higher, the air gets thinner, so the drag forces get relatively smaller.the drag forces get relatively smaller.


What we need is an equation that gives the What we need is an equation that gives the density of air as a function of altitude. density of air as a function of altitude.

There are several equations out there that There are several equations out there that I’ve seen, and I’m not sure which is truly I’ve seen, and I’m not sure which is truly correct- they all give fairly similar results.correct- they all give fairly similar results.


It may actually be a great exercise in critical It may actually be a great exercise in critical thinking for your class to discuss which of the thinking for your class to discuss which of the equations they find is “right”…equations they find is “right”…

For the sake of simplicity, I’ll use this one:For the sake of simplicity, I’ll use this one:

)7400/(225.1)( YeY

Level FiveLevel FiveTime Thrust Mass Weight Density Drag Net Force Acceleration Velocity Height

0 0.946 0.075 0.73575 1.225 0 0.21025 2.8033333 0 00.1 4.826 0.074 0.72758 1.225 1.2E-05 4.09841 55.259499 0.28033 0.01401670.2 9.936 0.073 0.7194 1.22495 0.00522 9.21138 125.60968 5.80628 0.31834750.3 14.09 0.073 0.71123 1.22475 0.05226 13.3265 183.814 18.3673 1.52702420.4 11.446 0.072 0.70305 1.22429 0.20912 10.5338 146.98361 36.7487 4.28281930.5 7.381 0.071 0.69488 1.22356 0.40962 6.2765 88.609449 51.447 8.69260250.6 6.151 0.07 0.6867 1.22264 0.56245 4.90185 70.026415 60.308 14.2803510.7 5.489 0.069 0.67853 1.22158 0.70005 4.11043 59.427859 67.3106 20.6612790.8 4.921 0.068 0.67035 1.22042 0.82833 3.42232 50.082718 73.2534 27.6894780.9 4.448 0.068 0.66218 1.21918 0.9445 2.84133 42.093704 78.2617 35.26523

1 4.258 0.067 0.654 1.21785 1.0477 2.5563 38.34457 82.471 43.3018641.1 4.542 0.066 0.64583 1.21646 1.14608 2.7501 41.773643 86.3055 51.7406891.2 4.164 0.065 0.63765 1.21501 1.2582 2.26815 34.89457 90.4828 60.5801061.3 4.448 0.064 0.62948 1.2135 1.35543 2.4631 38.385917 93.9723 69.8028641.4 4.448 0.063 0.6213 1.21193 1.46652 2.36018 37.265966 97.8109 79.3920241.5 4.448 0.063 0.61313 1.2103 1.57827 2.2566 36.105637 101.537 89.3594431.6 2.933 0.062 0.60495 1.20861 1.69015 0.6379 10.34433 105.148 99.6937211.7 1.325 0.061 0.59678 1.20688 1.72111 -0.99288 -16.321379 106.182 110.260251.8 0 0.06 0.5886 1.20517 1.66623 -2.25483 -37.580513 104.55 120.796891.9 0 0.06 0.5886 1.20349 1.54645 -2.13505 -35.584192 100.792 131.06402

2 0 0.06 0.5886 1.20189 1.43726 -2.02586 -33.764356 97.2339 140.96533


8 0 0.06 0.5886 1.16335 0.00142 -0.59002 -9.8336393 3.10469 382.083718.1 0 0.06 0.5886 1.16331 0.00066 -0.58926 -9.8210357 2.12133 382.345018.2 0 0.06 0.5886 1.16329 0.00019 -0.58879 -9.8131827 1.13922 382.508048.3 0 0.06 0.5886 1.16328 3.7E-06 -0.5886 -9.8100611 0.1579 382.572898.4 0 0.06 0.5886 1.16328 1E-04 -0.5887 -9.8116614 -0.8231 382.53964

Time Thrust Mass Weight Density Drag Net Force Acceleration Velocity Height

You’ll see that there is a difference of only You’ll see that there is a difference of only 4 meters for max height between Levels 4 meters for max height between Levels Four and Five.Four and Five.

My model rocket only went to 382 meters, My model rocket only went to 382 meters, where the air is still 95% the density of where the air is still 95% the density of ground level air- not a big difference, yet.ground level air- not a big difference, yet.


For model rockets, the change between Level For model rockets, the change between Level Four and Level Five is not very significant- but for Four and Level Five is not very significant- but for BIG rockets it becomes a BIG deal.BIG rockets it becomes a BIG deal.

At just over 3 miles up, the air density drops to At just over 3 miles up, the air density drops to 50% of that at ground level.50% of that at ground level.

That means that drag forces at that altitude would That means that drag forces at that altitude would be HALF of what they would be at ground level.be HALF of what they would be at ground level.

Level Five, SummaryLevel Five, Summary

As with the earlier stages, each refinement As with the earlier stages, each refinement we’ve made has made our simulation more we’ve made has made our simulation more accurate.accurate.

Level Five incorporated the correct weight Level Five incorporated the correct weight and thrust for my rocket- but used a and thrust for my rocket- but used a simplified representation of the drag force.simplified representation of the drag force.

Level Six will present a much more accurate Level Six will present a much more accurate drag force simulation.drag force simulation.

Level SixLevel Six

The coefficient of drag for a rocket The coefficient of drag for a rocket describes how aerodynamic it is.describes how aerodynamic it is.

It is a number between 0 and 1: It is a number between 0 and 1: A lower value means your design is very A lower value means your design is very

sleek (like a sports car)sleek (like a sports car) A higher value means your design isn’t very A higher value means your design isn’t very

streamlined (like a school bus)streamlined (like a school bus)

Level SixLevel Six

The goal of Level Six is to calculate the The goal of Level Six is to calculate the coefficient of drag for my rocket.coefficient of drag for my rocket.

Recall that we assigned an arbitrary value Recall that we assigned an arbitrary value of 0.56 for our previous calculations.of 0.56 for our previous calculations.

Unfortunately, Level Six will be the most Unfortunately, Level Six will be the most complicated math we’ve dealt with.complicated math we’ve dealt with.

Level SixLevel Six

What makes calculating the coefficient of drag so What makes calculating the coefficient of drag so complicated is that it depends not only on the complicated is that it depends not only on the interrelationship of size and shape of all the parts interrelationship of size and shape of all the parts of your rocket, but it also depends on the of your rocket, but it also depends on the velocity!velocity!

For most designs, it actually decreases as the For most designs, it actually decreases as the speed increases (until you approach the speed speed increases (until you approach the speed of sound, but our model rockets don’t even come of sound, but our model rockets don’t even come close to Mach 1, so we won’t worry about that.)close to Mach 1, so we won’t worry about that.)

Level SixLevel Six

We have several options:We have several options: We could hard-code the math into our spreadsheet We could hard-code the math into our spreadsheet

to calculate the coefficient of drag for each new to calculate the coefficient of drag for each new velocityvelocity

We could calculate the average coefficient of drag We could calculate the average coefficient of drag for the entire range of velocities and use that for the entire range of velocities and use that number in our calculations.number in our calculations.

We could place the rocket in a wind tunnel and find We could place the rocket in a wind tunnel and find the coefficient of drag for various velocities the coefficient of drag for various velocities experimentally.experimentally.

Level SixLevel Six

Aerodynamic drag on a rocket can be Aerodynamic drag on a rocket can be attributed to six main sources:attributed to six main sources: NoseNose BodyBody FinsFins TailTail Launch LugLaunch Lug InterferenceInterference

Level SixLevel Six

Walt Oler from Texas Tech University has Walt Oler from Texas Tech University has constructed an Excel spreadsheet that does constructed an Excel spreadsheet that does the ugly calculations for you.the ugly calculations for you.

You simply enter in the size and shape details You simply enter in the size and shape details for each component of your rocket.for each component of your rocket.

If he grants his permission, I’ll include the If he grants his permission, I’ll include the spreadsheet with this document.spreadsheet with this document.

a teacher’s guide to helping students simulate the flight of model rockets (version 2) kyle voge...

Documents