Rockets and 1/2 mv^2

[Janus]

A parabola or a hyperbola? I thought non bound objects follow a hyperbola - is this a case were the trajectory becomes a parabola?

 

At escape velocity it is a parabola, above it, it is a hyperbola. You could say that a parabola is a hyperbola with an eccentricity of exactly 1.

[Qfwfq]

Indeed, also as a conic curve the parabolas are the boundary between the other two classes, when the cutting plane is a parallel to a generator.

 

Max efficiency is at v=c where c is the exhaust speed and v is the speed of the airplane. This makes sense because moving a large amount of air very little is more efficient than moving a small amount of air a lot.

I would reckon that thrust would go to zero in that limit so that any fuel used may as well not be used. I call that zero efficiency and therefore suspect there must be a maximum efficiency at some v < c. I'll think about it when I can.

[enorbet2]

Greetz

Hopefully there is sufficient precedent in this thread for jets as well as rockets that this will not be too OT. Considering that jet engines are particularly efficient when the speed of the jet nears exhaust velocity, and further considering that Kelley Johnson of Skunkworks and other fame can obviously get as close as anyone so far can to top efficiency, I have to wonder what is the percentage of efficiency commonly achieved in high performance jet craft? By extension I have to wonder why the top speed of such jet craft is "secret".

 

The reason I ask this is that by such pictures as are commonly published like this one

http://www.cnw.mk.ua/weapons/airforce/razv/sr71/image/sr71ff.jpg

one can easily count the shock diamonds in the exhaust as at least 8 distinct diamonds visible. If I understand correctly, there appears one for each multiple of the speed of sound, so in that pic the exhaust velocity is roughly Mach 8.

 

Since the "official" speed for public consumption is Mac 3+ I wonder just who do they think they are kidding? Can't experts extrapolate the likely upper limits by an experience won percentage off exhaust velocity?

[modest]

Considering that jet engines are particularly efficient when the speed of the jet nears exhaust velocity, and further considering that Kelley Johnson of Skunkworks and other fame can obviously get as close as anyone so far can to top efficiency, I have to wonder what is the percentage of efficiency commonly achieved in high performance jet craft? By extension I have to wonder why the top speed of such jet craft is "secret".

 

The efficiency really wouldn't tell you the max velocity. If you knew the mass and the velocity of the exhaust and the drag force of the craft then you could, I suppose, get an estimate. I defer to greater minds.

 

The reason I ask this is that by such pictures as are commonly published like this one

http://www.cnw.mk.ua/weapons/airforce/razv/sr71/image/sr71ff.jpg

one can easily count the shock diamonds in the exhaust as at least 8 distinct diamonds visible. If I understand correctly, there appears one for each multiple of the speed of sound, so in that pic the exhaust velocity is roughly Mach 8.

 

Yeah... I don't think you can count shock diamonds to get an idea of the exhaust velocity. I'm pretty sure that's a reflection of how much over or under pressure the exhaust is to the atmosphere so that an underpressure exhaust would have more shock diamonds at sea level than at altitude with the same velocity. Where did you hear that you could count them to get a mach number in that way?

 

~modest

[enorbet2]

The efficiency really wouldn't tell you the max velocity. If you knew the mass and the velocity of the exhaust and the drag force of the craft then you could, I suppose, get an estimate. I defer to greater minds.

 

I apologize for not being clearer. I was thinking that "enemy" aircraft designers would be able to approximate drag coefficient, thrust, etc to get a pretty good idea of the performance of a known design even from a few pictures. Knowing the exhaust velocity would then give decent approximation of max speed. Also, considering older planes like the Bell X3 were touted as breaking 2K MPH (just under Mach 3) waaay back in early 60s and further considering advances, Mach 3 seems a silly cover up number.

 

 

Yeah... I don't think you can count shock diamonds to get an idea of the exhaust velocity. I'm pretty sure that's a reflection of how much over or under pressure the exhaust is to the atmosphere so that an underpressure exhaust would have more shock diamonds at sea level than at altitude with the same velocity. Where did you hear that you could count them to get a mach number in that way?

 

~modest

 

I read a lot of books about the early days of rocketry at White Sands first with captured

V2's, 2 stage V-2's and Viking rockets, when I was a kid. It was stated in a few of those books that shock diamonds represent Mach numbers. I will have to see if I can find any current data.

[CraigD]

The slingshot effect is a distinct thing, which doesn't require the rocket's thrust. Of course, you might ask where the energy comes from, without the fuel.

Bonus points to anyone who doesn’t already know the answer to this question who can figure it out without resorting to wiki/google/etc-ing it!

 

It’s one of those answers that seems glaringly obvious once you’ve seen it – but has profound consequences on the long-term evolution of the Solar System, and opens the door to a host of still unanswered astrophysical questions having little to do with rockets.

Ben took an elaborate shot at the question, prompting some elaborations, but what I imagined was much simpler.

 

The kinetic energy gained by a spacecraft from a gravity slingshot comes from the kinetic energy of the slingshotting planet.

 

For example, a maximum delta-V slingshot with Venus can, in principle, add about 70 km/s (twice Venus’s orbital speed, and more than the roughly 50 km/s Solar escape velocity for that location). For a typical 1000 kg spacecraft, the maneuver reduces the orbital speed of Venus by about 0.000001 m/s ([math]\sqrt{\frac{70000^2 \cdot 1000}{4.87 \times 10^{24}}}[/math]), about a 0.000000002% reduction. A gravity slingshot is a kind of elastic collision – not only momentum, but kinetic + potential energy is conserved.

 

The profound consequences of this dynamic is that the continuous slingshotting of small bodies by the planets changes their orbits. According to some theories, the stability of the solar system depends on this, with minor bodies acting as a sort of “pump”, increasing the orbit speed of some planets, while decreasing others, on a timescale of millions of years. The long-term structure and evolution of a stellar system appears to depend critically on both the large and small bodies in it, in a very complex, not fully understood way.

[Qfwfq]

The kinetic energy gained by a spacecraft from a gravity slingshot comes from the kinetic energy of the slingshotting planet.

Exactly.

 

As Janus said, it's an elastic collision. I one sits his arse, ehem, his coordinate choice, on Jupiter then one doesn't see the space probe gaining any kinetic energy. It follows a hyperbola, a symmetric trajectory. In the solar system coordinates, in which the astronomers are reckoning the overall itinerary, it's a wholly different kettle of fish.

 

Statistically, the rubble flying around the solar system definitely ought to act as a viscous fluid.

[enorbet2]

Greetz

Well I have wracked my brain and overworked Google and unfortunately I don't have a solid quote but I have some information that may help regarding shock diamonds as roughly an expression of exhaust velocity as relative to speed of sound ie: number of diamonds roughly equals the Mach multiplier.

 

While I fully understand that the diamonds are caused by overexpansion as explained here

Aerospaceweb.org | Ask Us - Shock Diamonds and Mach Disks and I include this quote

 

In an ideal gas, this process of expansion and contraction would continue on forever creating an infinite number of Mach disks. Real gases are not ideal, however, and friction created along the free jet boundary between the air and exhaust results in a turbulent shear layer. This layer creates a viscous damping that gradually dissipates the wave structure. This viscous friction eventually equalizes the pressure differences between the exhaust and ambient atmosphere so that the shock diamonds can no longer be formed.

 

It is my understanding that at a given atmospheric pressure, such as at sea level, this friction is more or less a constant and therefore the number of diamonds that form are in proportion to the exhaust velocity.

 

First of all nothing I have seen rules it out and they are indeed also called Mach Diamonds. The hardbound book from which I learned this rule of thumb (I read it in 1960) was written by and about the White Sands Proving Grounds scientists and their experiments with V2's, Aerobee Hi's, and especially the Viking rocket as seen here

The Viking Rocket

 

Please note that last picture on the right of Viking #9 shows seven (7) shock or mach diamonds. #14 reached a top speed of 4000 miles per hour at an altitude of 158 miles. Given even just a little loss for such an early rocket's inability to "catch up" to it's exhaust velocity the numbers are very close. However the only pic that had accompanying text displaying exhaust velocity instead of speed and altitude at brennschluss as well as diamonds, so far, is the picture at the bottom of the page located here

ch2-6

 

where it says

....the maximum exhaust velocity for the series was 3270 meters per second.* In general, performance with the regeneratively-cooled engine was considerably higher than that obtained with the water-cooled chambers. The experimenters attributed this not only to the elimination of heat losses, but also to a lower-density hydrogen entering the combustion chamber, which produced improved mixing and higher combustion efficiency. Figure 7 shows the regeneratively-cooled rocket operating in December 1949 during the series of tests.

 

note that the exhaust velocity listed is roughly Mach 9.6 but in that picture (figure 7) I can only count six, maybe seven mach diamonds. If can find a better quality pic I can try enhancing but I am not really expecting to see 2-3 more diamonds. That pic is only 12.6 KB.

 

Again I have yet to find any outright contradiction even though it is admittedly possible that what was assumed during the early days was erroneous. However it is also possible that such information has been dropped much like one is now lucky to get table salt in a modern kid's chemistry set, yet when I was coming up I had KNO3 and charcoal, Zinc dust and sulphur and numerous other chemicals that could easily make rocket fuel (which I did).

 

So the search continues.