Forest UCM Ch3 Rockets - Revision history

Foretony: /* Rocket problem 1 */

2014-09-13T13:59:17Z

Rocket problem 1

2014-09-13T13:38:50Z

2014-09-13T13:38:12Z

In terms of Thrust

2014-09-13T13:38:02Z

solving for the velocity

2014-09-13T13:35:46Z

solving for the velocity

2014-09-13T13:34:04Z

2014-09-13T13:31:52Z

2014-09-13T13:29:52Z

2014-09-13T13:26:44Z

2014-09-13T13:25:37Z

← Older revision		Revision as of 13:59, 13 September 2014
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		+	Remember the hobo problem? This is how rockets are propelled by expending fuel (mass).
		+
		+	Lets first think about the inverse rocket where mass is added instead of expelled.
		+
		+	Consider a railroad car with frictionless wheels moving at speed <math>v</math> on level ground.
		+
		+	A hobo handing from a tree drops down onto the railroad car as it is moving under the tree.
		+
		+	What is the final velocity of a railroad car after the hobo jumps on.
		+
		+	;Conservation of momentum
		+
		+	:<math>Mv = (M+m) v_f</math>
		+
		+	:<math>v_f = \frac{M}{M+m} v</math>
		+
		+	The railroad car slows down after the hobo jumps on.
		+
		+	Rockets work in a similar fashion. They speed up after the fuel "jumps" off.

	=Rocket Problem 2=		=Rocket Problem 2=

← Older revision		Revision as of 13:38, 13 September 2014
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		+	=Rocket problem 1=
		+
		+
		+
		+	=Rocket Problem 2=
	Consider a rocket of mass <math>m</math> moving at a speed <math>v</math> ejecting rocket fuel for propulsion.		Consider a rocket of mass <math>m</math> moving at a speed <math>v</math> ejecting rocket fuel for propulsion.

@@ Line 49: / Line 49: @@
 Then
-:<math>m dot{v} =</math> THRUST = Force on rocket as a result of expelling fuel
+:<math>m \dot{v} =</math> THRUST = Force on rocket as a result of expelling fuel
 [[Forest_UCM_MnAM#Rockets]]

← Older revision		Revision as of 13:38, 13 September 2014
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	where <math>m_0</math> is the initial Rocket mass		where <math>m_0</math> is the initial Rocket mass
		+
		+	==In terms of Thrust==
		+
		+	Defining
		+	:<math>\mbox{THRUST} = - \dot{m} u</math>
		+
		+	Then
		+
		+	:<math>m dot{v} =</math> THRUST = Force on rocket as a result of expelling fuel

	[[Forest_UCM_MnAM#Rockets]]		[[Forest_UCM_MnAM#Rockets]]

@@ Line 38: / Line 38: @@
 : <math>\int_{v_0}^v  = \int_{m_0}^m -u \frac{dm}{m}</math>
 : <math>v - v_0 = u \ln \left (\frac{m_0}{m} \right )</math>
 [[Forest_UCM_MnAM#Rockets]]

@@ Line 17: / Line 17: @@
 <math>V_FG =</math> Velocity of the fuel with respect to the ground
-;Conservation of Momentum
+= Apply Conservation of Momentum=
 :<math>mv = (-dm)V_{FG} + (m-(-dm))(v+dv)</math>
 :<math>dmV_{FG} = mdv + dm (v+dv)</math>
@@ Line 34: / Line 34: @@
 :<math>-udm  = mdv </math>
-;solving for the velocity
+==solving for the velocity==
 : <math>\int_{v_0}^v  = \int_{m_0}^m -u \frac{dm}{m}</math>
+: <math>v - v_0 = u \ln \left (\frac{m_0}{m} \right )</math>
 [[Forest_UCM_MnAM#Rockets]]

← Older revision		Revision as of 13:31, 13 September 2014
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	:<math>-udm = mdv </math>		:<math>-udm = mdv </math>

		+	;solving for the velocity
		+
		+	: <math>\int_{v_0}^v = \int_{m_0}^m -u \frac{dm}{m}</math>


	[[Forest_UCM_MnAM#Rockets]]		[[Forest_UCM_MnAM#Rockets]]

@@ Line 21: / Line 21: @@
 :<math>-dmV_{FG} = mdv + dm (v+dv)</math>
-The veocity of the fuel with respect to the ground may be written as the vector sum of the rocket velocith with respect to the ground and the velocity of the fuel with respect to the rocket (Galilean transormation)
+The velocity of the fuel with respect to the ground<math> (V_{FG})</math> may be written as the vector sum of the rocket's velocity with respect to the ground <math>(V_{RG})</math> and the velocity of the fuel with respect to the rocket <math>(V_{FR})</math> :Galilean transormation
 :<math>\vec{V}_{FG} - \vec{V}_{FR} + \vec{V}_{RG}</math>

← Older revision		Revision as of 13:25, 13 September 2014
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	<math>v+dv =v_{RG}</math> velocity of the rocket relative to the ground after ejecting fuel		<math>v+dv =v_{RG}</math> velocity of the rocket relative to the ground after ejecting fuel
		+
		+	<math>V_FG =</math> Velocity of the fuel with respect to the ground
		+
		+	;Conservation of Momentum
		+	:<math>mv = (-dm)V_{FG} + (m-(-dm))(v+dv)</math>
		+	:<math>-dmV_{FG} = mdv + dm (v+dv)</math>
		+
		+	The veocity of the fuel with respect to the ground may be written as the vector sum of the rocket velocith with respect to the ground and the velocity of the fuel with respect to the rocket (Galilean transormation)
		+
		+	:<math>\vec{V}_{FG} - \vec{V}_{FR} + \vec{V}_{RG}</math>



	[[Forest_UCM_MnAM#Rockets]]		[[Forest_UCM_MnAM#Rockets]]