en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q2370289

2013-03-14T09:28:34Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q2370289

← Older revision		Revision as of 11:28, 14 March 2013
Line 1:		Line 1:
	~~Oscar~~ is how he's ~~called~~ and ~~he totally loves this name~~. ~~He used to be unemployed but now he~~ is a ~~pc operator but his marketing by no means comes. One~~ of the ~~issues she enjoys most~~ is to ~~study comics~~ and ~~she'll be beginning some thing else along~~ with it. ~~South Dakota is where me~~ and ~~my spouse live~~.<br><br>~~Here~~ is ~~my blog~~: [~~http~~://Go.~~R2Om~~.~~com~~/~~dietmealdelivery74619~~ http://Go.~~R2Om~~.~~com~~/~~dietmealdelivery74619~~]		{{Astrodynamics \|Efficiency measures}}

			In [[aerospace engineering]], '''mass ratio''' is a measure of the efficiency of a [[rocket]]. It describes how much more massive the vehicle is with [[propellant]] than without; that is, the ratio of the rocket's ''wet mass'' (vehicle plus contents plus propellant) to its ''dry mass'' (vehicle plus contents). A more [[specific impulse\|efficient]] rocket design requires less propellant to achieve a given goal, and would therefore have a lower mass ratio; however, for any given efficiency a higher mass ratio typically permits the vehicle to achieve higher [[delta-v]].

			The mass ratio is a useful quantity for back-of-the-envelope rocketry calculations: it is an easy number to derive from either [[delta-v\|<math>\Delta v</math>]] or from rocket and propellant mass, and therefore serves as a handy bridge between the two. It is also a useful for getting an impression of the size of a rocket: while two rockets with [[propellant mass fraction\|mass fraction]]s of, say, 92% and 95% may appear similar, the corresponding mass ratios of 12.5 and 20 clearly indicate that the latter system requires much more propellant.

			Typical [[multistage rocket]]s have mass ratios in the range from 8 to 20. The [[Space Shuttle]], for example, has a mass ratio around 16.

			== Derivation ==

			The definition arises naturally from [[Tsiolkovsky's rocket equation]]:

			:<math>\Delta v = v_e \ln \frac {m_0} {m_1}</math>

			where
			:*[[delta-v\|Δ''v'']] is the desired change in the rocket's velocity
			:*''v<sub>e</sub>'' is the effective exhaust velocity (see [[specific impulse]])
			:*''m<sub>0</sub>'' is the initial mass (rocket plus contents plus propellant)
			:*''m<sub>1</sub>'' is the final mass (rocket plus contents)

			This equation can be rewritten in the following equivalent form:

			:<math>\frac {m_0} {m_1} = e ^ { \Delta v / v_e }</math>

			The fraction on the left-hand side of this equation is the rocket's mass ratio by definition.

			This equation indicates that a Δv of <math>n</math> times the exhaust velocity requires a mass ratio of <math>e^n</math>. For instance, for a vehicle to achieve a <math>\Delta v</math> of 2.5 times its exhaust velocity would require a mass ratio of <math>e^{2.5}</math> (approximately 12.2). One could say that a "velocity ratio" <!-- A term used in http://history.nasa.gov/SP-4404/app-a2.htm for instance --> of <math>n</math> requires a mass ratio of <math>e^n</math>.

			Sutton defines the mass ratio inversely as:<ref>Rocket Propulsion Elements, 7th Edition by George P. Sutton, Oscar Biblarz</ref>

			:<math>M_R = \frac {m_1} {m_0}</math>

			In this case, the values for mass fraction are always less than 1.

			== See also ==
			* [[Rocket fuel]]
			* [[Propellant mass fraction]]
			* [[Payload fraction]]

			== References ==
			{{cite book \| author=Zubrin, Robert \| title=Entering Space: Creating a Spacefaring Civilization \| publisher=Tarcher/Putnam \| year=1999 \| isbn=0-87477-975-8}}
			{{reflist}}

			[[Category:Astrodynamics]]
			[[Category:Mass]]
			[[Category:Ratios]]

en>Uni.Liu: /* Properties of finite morphisms */ Stein factorization is probably not enough. ZMT neither (at least the original version with normal variety as target space)

2012-08-19T21:28:35Z

Properties of finite morphisms: Stein factorization is probably not enough. ZMT neither (at least the original version with normal variety as target space)

New page

Oscar is how he's called and he totally loves this name. He used to be unemployed but now he is a pc operator but his marketing by no means comes. One of the issues she enjoys most is to study comics and she'll be beginning some thing else along with it. South Dakota is where me and my spouse live.<br><br>Here is my blog: [http://Go.R2Om.com/dietmealdelivery74619 http://Go.R2Om.com/dietmealdelivery74619]

Finite morphism - Revision history

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q2370289

en>Uni.Liu: /* Properties of finite morphisms */ Stein factorization is probably not enough. ZMT neither (at least the original version with normal variety as target space)