Rockets for Dummies

The Physics Behind Rockets

By:John Berrigan, berriganj@hdsb.ca

For:OAPT 2009, May 29th

Why space?

• Cool factor.• Canadian on ISS for 6 months…arrived today.• RMC contributions.• Who applied for the canadian Astronaut Program? • International competitions/milestones.• X-prizes• Return to moon

Why Rockets?

• Cool Factor• In the news• Lots of current applications (lots of great videos at the sites, be sure to check them out)

– Sub orbital tourism– Spacex (go to their website for cool videos)

– Virgin galactic– Xcor– Rocket Racing League– Armadillo (“doom” game creators rocket)– Blue Horizon (“Amazon” owner)– To name a few

The Physics Needed

• Conservation of momentum

• Impulse

• Newtons Laws– A GREAT Application for Newton’s 3rd law

• Electrostatics

• Magnetism

Conservation of Momentum

• From Newtons 3rd law(“prime” value represents the final value)

FAB = - FBA

mAaA = - mBaB

mAΔVA/ΔtA = - mBΔVB /ΔtB but Δt’s are the same so they cancel out

mA(VA’ – VA) = - mB(VB’ – VB) Rearranging,

mAVA+ mBVB = mAVA’+ mBVB’

pT = pT’ Total Momentum Before = Total Momentum After

Impulse

From Newton’s 2nd Law

FA = mAaA

FA = mAΔVA/ΔtA

Rearranging

FAΔtA = mAΔVA

FAΔtA = mA(VA’ –VA)

FAΔtA = mAVA’ – mAVA

FAΔtA = pA’ –pA

FAΔtA = ΔpA

Thrust equation

Impulse = FthrustΔt = Δp

Fthrust Δt = Δ(mV)Some calculus

FthrustΔt = (Δm)V + m(ΔV)From the frame of reference of the rocket engine, it is losing mass

and the velocity that it is throwing the mass out is constant, so….

Δm isn’t zero and ΔV is zero

FthrustΔt = (Δm)VOr

Fthrust = (Δm/Δt)V

Implications

What does the equation Fthrust = (Δm/Δt)V

imply?

To get the largest thrust you either want to throw the mass as fast as possible (large V), or throw out as much mass as possible in the shortest amount of time (large Δm/Δt).

What is better?

Conservation of momentumA 12 kg frictionless cart has 8 kg of “fuel” to

be thrown off the cart. The fuel can be thrown off at 10 m/s relative to the cart.

12 kg

8 kg

pT = mv = 20(0) = 0

pT’ = mv12 + mv8

= 12v12 + 8(v12-10) = 20v12 - 80

V12 - 10

12 kg

8 kg

v12

pT = pT’0 = 20v12 – 80V12 = 4

What if mass is thrown in 2 parts? In two 4 kg chunks!

pT = mv = 20(0) = 0

12 kg

4 4

pT’ = mv16 + mv4

= 16v16 + 4(v16-10) = 20v16 - 40

V16 - 10

12 kg

4

v16

4

pT = pT’0 = 20v16 – 40V16 = 2

So final velocity is faster if done in smaller parts!!

12 kg

4

pT = mv = 16(2) = 32

pT’ = mv12 + mv4

= 12v12 + 4(v12-10) = 16v12 - 40

V12 - 10

12 kg

4 kg

v12

pT = pT’32 = 16v12 – 40V12 = 72/16V12 = 4.5

What are the implications of this?

What is the smallest mass you can practically throw off?

1 g? 0.01 g? 1 ng?? Is an electron possible? Problem is there is a lot of mass left behind…So real answer is an atom.So the most efficient engine throws out one atom at a

time.This is called an ion engine!!

So What to do with this Info?

There are always pros and cons when doing things.

For throwing a smaller amount of mass out:

Pros:• you get much higher final velocity.

Cons:• Very low thrust. • Imagine throwing out one atom at a time..not

much force can be generated.

For Throwing lots of mass out at a time:Pro: • Higher thrustCon:• Slower final velocity If you want to get out of a “gravity well”. You

want high thrust so you throw lots of mass out at a time.

If you want to go faster, low thrust is fine.Efficiency is key for this!!!

Specific Impulse (Isp)

We need to introduce a way to deal with how efficient a rocket engine design is.

Specific impulse is a way to do this.

By definition,

Isp is the length of time 1 kg of fuel can produce 9.8 N of thrust.

Isp

Isp actually makes some sense.

If one engine has an Isp of 3 s and the other 300 s which one is better?

If it takes 3 s to produce the 9.8 N of thrust you are using the fuel very quickly.

If it takes 300 s to produce the 9.8 N of thrust, you are taking a long time to use the mass and it is therefore much more efficient.

Applications

How do we use the Isp formula and thrust formulas?

Example

If a rocket engine has an Isp of 300 s, how much fuel must be used to produce 1000N of thrust?

Isp

FΔt = Δm(v)

9.8(300) = 1v

V = 2940 m/s

Thrust

F = (Δm/Δt)(v)

1000 = (Δm/Δt)(2940)

Δm/Δt = 0.340 kg/s

Combining formulas

If we combine the thrust and ISP formulas we get the equation,

Fthrust = (Δm/Δt)(Isp*9.8)

So why don’t we use high ISP rocket engines all the time?

Has to do with the physics and thrust to mass ratio.

Rocket Engine design

Three main parts1. You need something the “throw”. Which

is the fuel.2. You need Energy. Used to throw the

fuel.3. You need some sort of nozzle. Used to

direct the fuel in the opposite direction you want to go.

Energy

So where can we get the energy from?

Easiest to get is from chemical energy. So you need to have 2 substances. A fuel and an oxidizer.

When the two substances react energy is released.

FuelWhat “fuel” should we use? If we use a chemical rocket we need to use chemicals

that allows for a lot of energy to be released when they burn, but at the same time be safe to handle.

Liquid oxygen and hydrogen is the best combo in terms of efficiency. (Isp ~ 400 s)

Lots of energy is released as well as small molecular mass, 18 amu, being thrown out the back end but hard to develop due to cryogenic issues and large tankage mass due to low density of H2.

Nozzle

We want the fuel thrown out the back. Inside the chamber two chemicals react.

So here a series of a simple motor designs. Why is the next one better than the previous?

thrust

thrust

Smaller opening, higher pressure. Using PV = nRT, means higher temperature.Higher temp means higher velocity of particles.

Why not make the hole really tiny?Rocket engine will blow up or be too heavy to lift with the thrust produced.

thrust

Nozzle redirects the released particles in a backward direction to increase the forward thrust.

Why not make the nozzle even bigger?

2 reasons:1. There is a mass penalty, a bigger nozzle can have a

detrimental over all effect.2. If in atmosphere, Expansion issues involving the gas in the

nozzle.As the gas expands in the nozzle, its pressure decreases. So the

gas can actually get below the atmospheric conditions so the atmosphere can leak into the nozzle increasing drag on the rocket.

If the bell is to small the gas can leak out of the nozzle lowering over all thrust. So the nozzle should be designed for the atmospheric conditions it is to be launched in.

Nozzle design

Ideal rocket exhaust

Under expanded rocket exhaust

Over expanded rocket exhaust

Other Chemical Fuel sources?

Solid motors:– Combines fuel and oxidizer.. – Simplest rocket engine but low ISP. Can’t be shut off (Isp =

270 s)

Other fuels:– Rubber/Nitrous oxide. Low ISP, “safer” do to non

cryogenic sources of fuel. (Isp = 235 s)– Oxygen/Kerosene. Reasonable ISP, smaller tankage mass.

(Isp = 275 s)– Hydrazine (Isp = 220 s) Nasty stuff…Benefit Long storage

Nitrous motor testFrom yesterdays press release

Other Energy sourcesThe one thing that you need is to some how get the

molecules of the fuel moving fast out the back of the rocket. So moving fast means high temperature.

Chemical rockets have a limited amount of energy that can be released during the reaction.

So alternative is using a nuclear reactor and use the fuel, preferably H2 due to low a.m.u. or even water which is easier to store, as a coolant. The hot “fuel”/Coolant is then thrown out the back.

How fast is dependent upon “reactor” design. Isp is around 800 s

ElectrostaticsA nuclear engine is restricted by the materials used in

the reactor. Don’t want it to melt. Another way is speed up charged particles is a charged

plate. So you ionize the particles and send them in between a charged plate.

Use solar cells for the power to ionize the atoms and to create the voltage for the plates.

This is an ion engine. Limited power though. Isp = 3000 s.

Ion.swf

Ion EngineBiggest problem is wear and tear on the electrodes. They have been used for EXTENDED periods on the deep space

1 mission, and currently on the DAWN mission.Some satellites are now using ion engines for orbit adjustments.New engine design called VASIMR. Variable Specific Impulse Magnetic Rocket. VERY cool Technology. Best to use Nuclear reactor as a source

of energy for greater power/Thrust. Can adjust the Magnetic field/Heating to change flow of ions.

So high thrust or low thrust.

Isp theoretically ranges from 3 000s to 30 000 s.

VASIMR EngineNothing in the way of the emitted ions. Almost no wear and tear like

other engine designs.Notice the “nozzle” for this engine is the magnetic field lines..VERY

cool!!

ISP Curve

Images from

http://www.adastrarocket.com/VASIMR.html

Rocket engine design

In General,

High Isp is low thrust,

Low ISP can have high thrust.

Currently, low Isp high thrust engines get us into orbit. Ideally a high ISP low thrust move us around.

Comparison

Thrust to mass ratio

So to make the best rocket you need to look at the economics.

A highly efficient engine may just weigh too much. So a lower efficient engine may be better. Example:A typical Ion engine of thrust on 0.1 N has no chance

of lifting itself from the surface of the earth so no point using it.

But in space it can easily be used to increase its speed.

“Famous” Rockets

Saturn 5

1st stage

Lox/Ker

Other stages

Lox/LH

Space Shuttle

1st stage/2nd stage

Solid rocket

Lox/LH

“3rd stage”

Hydrazine..

NASTY stuff but VERY reliable

Soyuz

All 3 stages

Lox/Ker

“Space Ship one”

One the $10 million dollar X-prize.

1st stage

N2O2/“rubber”

Falcon 1 1st and 2nd stages

Lox/Ker

Falcon 9

Dawn Spacecraft

Ion Engine

385 kg of Xenon

For many YEARS of firing

Tie Fighter

Twin Ion Engine

EnterpriseWarp Drive and Impulse Engines