What equation governs the relationship between the mass and velocity of exhaust shot out the back of the rocket and the velocity of the rocket? m1v1 = m2v2?
Thanks,
Rob
Yes, that’s correct. Momentum is conserved, so the change in velocity of the rocket is dependent on the change in velocity of the exhaust according to the equation you posted.
Momentum is conserved, but it’s easy to get confused when dealing with rockets. My freshman physics book took a careful step-by-step in deriving the Rocket Equation. It’s easy to get lost in looking at what velocities are measured in relation to. Simply differentiating can get you in trouble.
I mean, it is rocket science…
The quoted equation is pretty corrupted in the post above. Go to the cited article, or here:
Bear in mind that m2 is non-linear (it decreases as m1 is pushed out the back). Hence, m1v1 must exceed m2v2 or the rocket will fail to accelerate. Failing to adjust for this results in the “pogo effect”, where the rocket’s v2 drops to adjust to having overrun the thrust.
I’m sorry, but none of the above is strictly correct. Any transfer of momentum to mass outside of the vehicle with an ejection (exhaust) velocity vector opposing vehicle motion from the vehicle reference frame will result in some degree of acceleration per the “rocket equation”.
The equation that governs this is referred to as the Tsiolkovsky rocket equation after its publisher Russian mathematician and father of astronautic science Konstantin Tsiolkovsky, and is ∆V = V[sub]e[/sub] × ln(m[sub]i[/sub]/m[sub]f[/sub]). While not obvious without a fair amount of algebraic manipulation, this is actually just a restatement of the conservation of momentum relationship. This well explained at Robert Braeuing’s Rocket and Space Technology website, which is IMHO the best Internet resource for information about the fundamentals of rocket propulsion.
The tyranny of this equation does dictate that except for unreasonably high exhaust velocities most of the momentum transferred from ejected propellant is just used to accelerate propellant to be used later in action time, but that’s the physics and the reason we use jet turbines (which draw most of their propellant and all oxidizer out of the ambient air) rather than rockets to travel quickly around the planet, and also one of the various reasons why it is not trivial to go to Mars or any other planet, and while we will not be traveling to other star systems using any conventional propulsive technology.
POGO (which is actually not an acronym) is a specific effect that occurs in the feed system of high pressure liquid propellant rockets and is caused by the interaction of structural resonance of the feed system with the pressure head at the turbine inlet. It has nothing to do with external momentum transfer except that it does result in modal variations in thrust, but the cause is entirely self-induced and has nothing to do with external sources (aerodynamic drag, plume interaction, aeroacoustic vibration, gravity, et cetera). POGO can be reduced or eliminated with good design practices such as decoupling structural modal response and propellant feed response and using fluid accumulators to dampen any instabilities in the feed system.
Stranger
Wait a minute, I did get the non-linear mass part right: the vehicle does lose mass as it exhausts fuel, you cannot deny that. That means the effective thrust increases because the rocket’s mass is decreasing (as you said, the tyranny of having to push the fuel).
But thanks for straightening me out on the POGO effect.
Another very interesting thing to notice in these equations is that the speed gained by the rocket is more when the exhaust particles have small mass but higher velocity.
So say mv (momentum) of the exhaust particle is constant, then a smaller mass pushed out at a higher velocity is better than a bigger mass pushed out at a smaller velocity.
This is the reason ion thrusters work better : Ion thruster - Wikipedia
The thrust at a given mass flow rate and exhaust velocity remains the same. The resultant acceleration experienced by the spacecraft may increase because there is less mass to push, but this is “non-linear” only in the sense that the velocity is related to the natural log of the ratio of expended mass to total initial mass; the propellant mass efficiency (also known as specific impulse or I[SUB]sp[/SUB]) remains the same and is essentially controlled by the mass flow rate and thermodynamic properties of the expelled propellant as it passes the exit plane of the nozzle.
“Work better” can be assessed only in the context of the application. It is true that (most) ion thrusters have a significantly better specific impulse than chemically-powered thrusters; however, because much of the energy put into the system is retained as thermal energy by the expelled propellant rather than converted into momentum in the chamber or by expansion through the exit nozzle, their energy efficiency is shockingly low. The necessary engine mass and the required thermal protection to prevent erosion or material breakdown (usually requiring a thick layer of ceramic or quartz in the chamber) also means that the thrust-to-inert-mass ratio for most ion thrusters is very low; too low for terrestrial to orbit launch, and even so low that most types of thrusters are only suited for station-holding applications where a low continuous thrust with very slow impulse maneuvers over a long duration is acceptable. For larger thrusters at high thrust levels, some means of carrying away waste heat is mandatory, which requires a large attendant cooling and radiator system that adds significantly to the inert mass of the spacecraft as does the need for either an internal high density power supply or a giant collector array for solar electric propulsion (see Mission to Mars using ‘Six Not So Easy’ Pieces, M. Raftery).
Stranger
That pdf is slightly non-useful, it looks like 84 pages of presentation slides without the commentary.
Yes, that was a briefing to the Future In-Space Operations Colloquium chaired by NASA GSFC. I didn’t provide it as a comprehensive discussion of crewed Mars mission operations (and given its presumption of ISS utilization for such a mission I have some disagreement with the approach); it was merely to give a notion for the size of a solar electric powered ion propulsion vehicle (the SEP Tug). If you look in the same directory you’ll find additional papers that describe the proposed system architecture at a very high level without much in the way of technical details, which don’t exist at this point anyway.
Stranger
Thanks Stranger - good information. It was my intention to point out to the OP that hurling away a 2 oz golf ball at 200 mph is better for propulsion than hurling away a 5 oz baseball at 80 mph - although the momentum of both are the same. Can you help me illustrate this in a succinct manner ?
E[sub]k[/sub]=[sup]1[/sup]/[sub]2[/sub]mv[sup]2[/sup]
So there, I think, would be your explanation: the speed factor of the kinetic energy algorithm is squared.
Yes, but first, let’s convert your example into sensible (e.g. metric) units and assume a “standard” astronaut at m[SUB]a[/SUB] = 80 kg.:
A. An astronaut hurling: “a 2 oz golf ball at 200 mph” ==
m[SUB]a[/SUB] = 80 kg
m[SUB]p[/SUB] = 0.057 kg
v[SUB]e[/SUB] = 89.4 m/s
B. An astronaut hurling: “a 5 oz baseball at 80 mph” ==
m[SUB]a[/SUB] = 80 kg
m[SUB]p[/SUB] = 0.142 kg
v[SUB]e[/SUB] = 35.8 m/s
Taking the rocket equation as stated above and rearranging slightly to separate inert mass (m[SUB]a[/SUB], i.e the astronaut) and propellant mass (m[SUB]p[/SUB], i.e the ball), we get
∆v = v[sub]e[/sub] × ln(m[sub]i[/sub]/m[sub]f[/sub]) = v[sub]e[/sub] × ln((m[sub]a[/sub]+m[sub]p[/sub])/m[sub]a[/sub])
For a single thrown ball, we get the following results:
∆v[sub]A[/sub] = 0.06375 m/s
∆v[sub]b[/sub] = 0.06349 m/s
Not too much difference, and in fact if we rounded to significant figures it would be lost in the noise.
Now lets look at an astronaut throwing 100 balls (i.e. m[sub]p[/sub] is multiplied by a factor or 100):
∆v[sub]A[/sub] = 6.15 m/s
∆v[sub]b[/sub] = 5.85 m/s
The resulting difference is numerically significant as Astronaut B achieves only 95% of the ∆v compared to Astronaut A. I bet B wishes he’d taken up golf now.
Now lets look at an astronaut throwing 10 000 balls (i.e. m[sub]p[/sub] is multiplied by a factor or 10[SUP]4[/SUP]):
∆v[sub]A[/sub] = 187 m/s
∆v[sub]b[/sub] = 105 m/s
Astronaut B only gets 56% of the ∆v that Astronaut A gets, which means that A gets to the airlock first, cycles through, fires up the particle beam cannon, and turns B into a cloud of highly dissociated plasma. (In my imagination, of course…this is how I cope with the mundanity of everyday terrestrial life and the boredom of system requirements reviews and configuration change boards.)
The reason that there are differences is because of the effect on velocity of much smaller individual propellant masses against the overall mass of the vehicle (or in this case, astronaut) being propellant, including the residual propellant. Breaking the propellant into smaller chunks which are being ejected with more speed gives a slightly greater resultant velocity per unit mass that is compounded over the total mass of the propellant. Or think of it another way; you’ll enjoy drinking 18 shots of Glenmorangie Single Malt Quinta Ruban spread out evenly over a period of three days rather than guzzling half a pint of the same at three sittings. Trust me, this is the absolutely the case; you don’t need to run the experiment for yourself unless your name is Sterling Archer and you have to keep your buzz going to complete your mission to rescue an ocelot on top of a moving train while being alternately attacked by real and fake Canadian Mounties.
Note that it really wasn’t necessary to convert units as the masses cancel each other out inside the natural logarithm, but it gave me larger numbers to work with, and all real engineering should either be done in consistent base or normalized units, not arbitrary unit systems based on the length of someone’s appendage.
Not to be overly harsh, but please stop guessing at answers; it just adds to the confusion of what should be a straightforward explanation of basic mechanics. Conversion of thermodynamic (scalar) energy to directional kinetic (vector) energy does have a place in assessing the overall performance and efficiency of a propulsion system but plays no part in comparing the propellant mass efficiency of the system at a given impulse or exhaust velocity.
Stranger
I would quibble with that statement. Take the NSTAR engine from the Deep Space 1 mission: 3200 s Isp, and 92 mN thrust at a 2.3 KW power level.
That Isp corresponds to a 31,392 m/s exit velocity, and with the thrust level gives a 2.931e-6 kg/s mass flow rate. One second of thrust will thus have 1444 J of KE, which means our effective KE power is 1444 W.
That’s more than half the input power, which really isn’t so bad. Most of the input power actually does go into KE of the exhaust. At most, only 856 W go into raising the exhaust temperature (and presumably some of this is retained by the thruster itself).
The main reason why ion thrusters–and in fact all high-Isp thrusters–have shockingly low energy efficiency is simply because the exhaust is at high speed and thus has high KE. Even an infinitely efficient high-Isp thruster (say, a superconducting mass driver) would have very low energy efficiency compared to a chemical rocket. The advantage of course is in propellant efficiency.
You have to also remember momentum = impulse. Where Momentum = MV and Impulse = Fdt such that it is true that MV = Fdt. Although it has been explained to me in this way, Fdt seemed to have only practical use in strictly academic applications. Like, how does one measure the miniscule dt in which a ball and bat are in contact?
Just…stop. Look at a basic physics textbook. Figure out what you are trying to say before writing things that are just not correct and that are going to confuse everyone else.
Impulse is the integral of applied force with time. In other words, you need to sum or integrate the force over the period to find the total impulse. Specific impulse is the total impulse divided by the weight of expended propellant and is determined from flight or test data, although an estimate can be made based upon the anticipated thrust profile. (Some measures use mass instead of weight, which is actually more sensible but not the industry standard in propulsion engineering for historical reasons.)
The impulse over a very short period, e.g. an impact transfer is developed as an acceleration. In terms of its structural and modal effects it is developed as a shock response spectrum (SRS) which separates the effects as various peak accelerations of an idealized single degree of freedom (1DOF) damped spring system at frequencies spaced along even logarithmic intervals.
Stranger
It seems that Newton’s laws were still controversial in the 20th century:
[QUOTE=Topics of The Times, Jan. 13, 1920 in The New York Times]
That Professor Goddard, with his ‘chair’ in Clark College and the countenancing of the Smithsonian Institution, does not know the relation of action to reaction, and of the need to have something better than a vacuum against which to react – to say that would be absurd. Of course he only seems to lack the knowledge ladled out daily in high schools.
[/QUOTE]
(N.Y. Times posted a retraction 17 July, 1969 – shortly before the first Moon landing.)
Wow - the smugness in that 1920 reference, it burns.
Was that a letter to the Editor from one of those types you can still see today, Dunning-Krugering their way through life, or was it a journalist or some sort of expert commentator? The retraction implies that it was a journalist (perhaps an editorialist) but it is not clear.
N.Y. Times archives – over a century! – used to be freely available; now a paid subscription is required. (Searching and synopsis is still free, and often good enough.)