http://wiki.newmars.com/api.php?action=feedcontributions&user=Techer&feedformat=atom NewMarsWiki - User contributions [en] 2022-06-29T13:44:58Z User contributions MediaWiki 1.29.1 http://wiki.newmars.com/index.php?title=Rocket_equation&diff=892 Rocket equation 2010-04-09T21:04:28Z <p>Techer: /* Relativistic Rocket Equation */</p> <hr /> <div>The rocket equation is one solution to the momentum method of calculating a rocket's [[delta v]]. The equation is:<br /> <br /> v&lt;sub&gt;f&lt;/sub&gt; - v&lt;sub&gt;i&lt;/sub&gt; = v&lt;sub&gt;e&lt;/sub&gt; ''ln''R<br /> <br /> where: <br /> <br /> &lt;math&gt;\Delta&lt;/math&gt;v = v&lt;sub&gt;f&lt;/sub&gt; - v&lt;sub&gt;i&lt;/sub&gt;<br /> v&lt;sub&gt;f&lt;/sub&gt; = Final Velocity of Rocket<br /> v&lt;sub&gt;i&lt;/sub&gt; = Initial Velocity of Rocket<br /> v&lt;sub&gt;e&lt;/sub&gt; = [[Exhaust Velocity]] of the rocket<br /> R = [[Mass Ratio]] of Rocket<br /> <br /> ''ln''R is the natural logarithm of R.<br /> <br /> ==Mathematics==<br /> <br /> The rocket equation is derived from the conservation of momentum.<br /> <br /> &lt;math&gt;\Sigma p_{i} = \Sigma p_{f} = Constant &lt;/math&gt;&lt;br&gt;<br /> <br /> If mass is exhausted from a rocket engine at some mass flow rate m'&lt;sub&gt;e&lt;/sub&gt; and some exhaust velocity v&lt;sub&gt;e&lt;/sub&gt;, then that exhaust steam carries momentum, manifested as [[thrust]]:<br /> <br /> &lt;math&gt;{dp \over dt} = m'_{e} v_{e} = v_{e} {dm \over dt} = m {dv \over dt}&lt;/math&gt;&lt;br&gt;<br /> <br /> Using the algebraic representation of derivatives, this differential equation can be reduced to:<br /> <br /> &lt;math&gt;dv = v_{e} {dm \over m}&lt;/math&gt;&lt;br&gt;<br /> <br /> If the exhaust velocity is constant, then the integral of this is the rocket equation.<br /> <br /> &lt;math&gt;\Delta v = \int dv = v_{e} \int {dm \over m} = v_{e} ln (R)&lt;/math&gt;&lt;br&gt;<br /> <br /> Conservation of momentum can be applied to more propulsion situations than the simple case presented above. For example, under the influence of constant gravity:<br /> <br /> &lt;math&gt;{dp \over dt} = m'_{e} v_{e} + g cos( \theta )&lt;/math&gt;&lt;br&gt;<br /> <br /> Using the substitutions listed above, the integral of this relationship becomes:<br /> <br /> &lt;math&gt;\Delta v = v_{e} ln (R) + g t cos( /theta )&lt;/math&gt;&lt;br&gt;<br /> <br /> If the exhaust consists of two separate gas streams whose mass flow rates keep some constant ratio, r, to each other:<br /> <br /> &lt;math&gt;r = {m'_{1} \over m'_{2}}&lt;/math&gt;&lt;br&gt;<br /> <br /> &lt;math&gt;v_{avg} = { v_{1} m'_{1} + v_{2} m'_{2} \over m'_{1} + m'_{2}}&lt;/math&gt;&lt;br&gt;<br /> <br /> Thus:<br /> <br /> &lt;math&gt;v_{avg} = { r v_{1} + v_{2} \over r + 1 }&lt;/math&gt;<br /> <br /> and v&lt;sub&gt;avg&lt;/sub&gt; can be substituted into the rocket equation rather than v&lt;sub&gt;e&lt;/sub&gt;.<br /> <br /> ==Example==<br /> <br /> ==Relativistic Rocket Equation==<br /> The rocket equation for speeds that become relativistic (~0.1c) is: <br /> <br /> &lt;math&gt;\frac{\Delta v}{c} = \tanh(\frac{v_e \ln(m_i/m_f)}{c\sqrt{1-\frac{v_e^2}{c^2}}})&lt;/math&gt;<br /> <br /> ''tanh'' is the hyperbolic tangent function.<br /> <br /> ==External Links==<br /> <br /> *[http://en.wikipedia.org/wiki/Rocket_equation Wikipedia Article: Tsiolkovsky Rocket Equation]</div> Techer http://wiki.newmars.com/index.php?title=Rocket_equation&diff=803 Rocket equation 2009-10-22T23:23:38Z <p>Techer: </p> <hr /> <div>The rocket equation is one solution to the momentum method of calculating a rocket's [[delta v]]. The equation is:<br /> <br /> v&lt;sub&gt;f&lt;/sub&gt; - v&lt;sub&gt;i&lt;/sub&gt; = v&lt;sub&gt;e&lt;/sub&gt; ''ln''R<br /> <br /> where: <br /> <br /> &lt;math&gt;\Delta&lt;/math&gt;v = v&lt;sub&gt;f&lt;/sub&gt; - v&lt;sub&gt;i&lt;/sub&gt;<br /> v&lt;sub&gt;f&lt;/sub&gt; = Final Velocity of Rocket<br /> v&lt;sub&gt;i&lt;/sub&gt; = Initial Velocity of Rocket<br /> v&lt;sub&gt;e&lt;/sub&gt; = [[Exhaust Velocity]] of the rocket<br /> R = [[Mass Ratio]] of Rocket<br /> <br /> ''ln''R is the natural logarithm of R.<br /> <br /> ==Mathematics==<br /> <br /> The rocket equation is derived from the conservation of momentum.<br /> <br /> &lt;math&gt;\Sigma p_{i} = \Sigma p_{f} = Constant &lt;/math&gt;&lt;br&gt;<br /> <br /> If mass is exhausted from a rocket engine at some mass flow rate m'&lt;sub&gt;e&lt;/sub&gt; and some exhaust velocity v&lt;sub&gt;e&lt;/sub&gt;, then that exhaust steam carries momentum, manifested as [[thrust]]:<br /> <br /> &lt;math&gt;{dp \over dt} = m'_{e} v_{e} = v_{e} {dm \over dt} = m {dv \over dt}&lt;/math&gt;&lt;br&gt;<br /> <br /> Using the algebraic representation of derivatives, this differential equation can be reduced to:<br /> <br /> &lt;math&gt;dv = v_{e} {dm \over m}&lt;/math&gt;&lt;br&gt;<br /> <br /> If the exhaust velocity is constant, then the integral of this is the rocket equation.<br /> <br /> &lt;math&gt;\Delta v = \int dv = v_{e} \int {dm \over m} = v_{e} ln (R)&lt;/math&gt;&lt;br&gt;<br /> <br /> Conservation of momentum can be applied to more propulsion situations than the simple case presented above. For example, under the influence of constant gravity:<br /> <br /> &lt;math&gt;{dp \over dt} = m'_{e} v_{e} + g cos( \theta )&lt;/math&gt;&lt;br&gt;<br /> <br /> Using the substitutions listed above, the integral of this relationship becomes:<br /> <br /> &lt;math&gt;\Delta v = v_{e} ln (R) + g t cos( /theta )&lt;/math&gt;&lt;br&gt;<br /> <br /> If the exhaust consists of two separate gas streams whose mass flow rates keep some constant ratio, r, to each other:<br /> <br /> &lt;math&gt;r = {m'_{1} \over m'_{2}}&lt;/math&gt;&lt;br&gt;<br /> <br /> &lt;math&gt;v_{avg} = { v_{1} m'_{1} + v_{2} m'_{2} \over m'_{1} + m'_{2}}&lt;/math&gt;&lt;br&gt;<br /> <br /> Thus:<br /> <br /> &lt;math&gt;v_{avg} = { r v_{1} + v_{2} \over r + 1 }&lt;/math&gt;<br /> <br /> and v&lt;sub&gt;avg&lt;/sub&gt; can be substituted into the rocket equation rather than v&lt;sub&gt;e&lt;/sub&gt;.<br /> <br /> ==Example==<br /> <br /> ==Relativistic Rocket Equation==<br /> The rocket equation for speeds that become relativistic (~0.1c) is: <br /> <br /> &lt;math&gt;\Delta&lt;/math&gt;v \over c = ''tanh'' ( {v&lt;sub&gt;e&lt;/sub&gt; \over c} ''ln'' R)<br /> <br /> ''tanh'' is the hyperbolic tangent function.<br /> <br /> <br /> <br /> <br /> ==External Links==<br /> <br /> *[http://en.wikipedia.org/wiki/Rocket_equation Wikipedia Article: Tsiolkovsky Rocket Equation]</div> Techer http://wiki.newmars.com/index.php?title=Rocket_equation&diff=802 Rocket equation 2009-10-18T21:44:40Z <p>Techer: </p> <hr /> <div>The rocket equation is one solution to the momentum method of calculating a rocket's [[delta v]]. The equation is:<br /> <br /> v&lt;sub&gt;f&lt;/sub&gt; - v&lt;sub&gt;i&lt;/sub&gt; = v&lt;sub&gt;e&lt;/sub&gt; ''ln''R<br /> <br /> where: <br /> <br /> &lt;math&gt;\Delta&lt;/math&gt;v = v&lt;sub&gt;f&lt;/sub&gt; - v&lt;sub&gt;i&lt;/sub&gt;<br /> v&lt;sub&gt;f&lt;/sub&gt; = Final Velocity of Rocket<br /> v&lt;sub&gt;i&lt;/sub&gt; = Initial Velocity of Rocket<br /> v&lt;sub&gt;e&lt;/sub&gt; = [[Exhaust Velocity]] of the rocket<br /> R = [[Mass Ratio]] of Rocket<br /> <br /> ''ln''R is the natural logarithm of R.<br /> <br /> ==Mathematics==<br /> <br /> The rocket equation is derived from the conservation of momentum.<br /> <br /> &lt;math&gt;\Sigma p_{i} = \Sigma p_{f} = Constant &lt;/math&gt;&lt;br&gt;<br /> <br /> If mass is exhausted from a rocket engine at some mass flow rate m'&lt;sub&gt;e&lt;/sub&gt; and some exhaust velocity v&lt;sub&gt;e&lt;/sub&gt;, then that exhaust steam carries momentum, manifested as [[thrust]]:<br /> <br /> &lt;math&gt;{dp \over dt} = m'_{e} v_{e} = v_{e} {dm \over dt} = m {dv \over dt}&lt;/math&gt;&lt;br&gt;<br /> <br /> Using the algebraic representation of derivatives, this differential equation can be reduced to:<br /> <br /> &lt;math&gt;dv = v_{e} {dm \over m}&lt;/math&gt;&lt;br&gt;<br /> <br /> If the exhaust velocity is constant, then the integral of this is the rocket equation.<br /> <br /> &lt;math&gt;\Delta v = \int dv = v_{e} \int {dm \over m} = v_{e} ln (R)&lt;/math&gt;&lt;br&gt;<br /> <br /> Conservation of momentum can be applied to more propulsion situations than the simple case presented above. For example, under the influence of constant gravity:<br /> <br /> &lt;math&gt;{dp \over dt} = m'_{e} v_{e} + g cos( \theta )&lt;/math&gt;&lt;br&gt;<br /> <br /> Using the substitutions listed above, the integral of this relationship becomes:<br /> <br /> &lt;math&gt;\Delta v = v_{e} ln (R) + g t cos( /theta )&lt;/math&gt;&lt;br&gt;<br /> <br /> If the exhaust consists of two separate gas streams whose mass flow rates keep some constant ratio, r, to each other:<br /> <br /> &lt;math&gt;r = {m'_{1} \over m'_{2}}&lt;/math&gt;&lt;br&gt;<br /> <br /> &lt;math&gt;v_{avg} = { v_{1} m'_{1} + v_{2} m'_{2} \over m'_{1} + m'_{2}}&lt;/math&gt;&lt;br&gt;<br /> <br /> Thus:<br /> <br /> &lt;math&gt;v_{avg} = { r v_{1} + v_{2} \over r + 1 }&lt;/math&gt;<br /> <br /> and v&lt;sub&gt;avg&lt;/sub&gt; can be substituted into the rocket equation rather than v&lt;sub&gt;e&lt;/sub&gt;.<br /> <br /> ==Example==<br /> <br /> ==Relativistic Rocket Equation==<br /> The rocket equation for speeds that become relativistic (~0.1c) is: <br /> <br /> &lt;math&gt;\Delta&lt;/math&gt;v \over c = ''tanh'' ( ''ln'' R)<br /> <br /> ''tanh'' is the hyperbolic tangent function.<br /> <br /> <br /> <br /> <br /> ==External Links==<br /> <br /> *[http://en.wikipedia.org/wiki/Rocket_equation Wikipedia Article: Tsiolkovsky Rocket Equation]</div> Techer