Forest UCM Ch3 Rockets

Consider a rocket of mass [math]\displaystyle{ m }[/math] moving at a speed [math]\displaystyle{ v }[/math] ejecting rocket fuel for propulsion.

File:Forest UCM Ch3 Rockets Fig.xfig.txt

[math]\displaystyle{ m = }[/math] mass of Fuel + Rocket

[math]\displaystyle{ -dm = }[/math] mass of Fuel ejected over time interval [math]\displaystyle{ dt }[/math] ( [math]\displaystyle{ dm }[/math] = mass lost by rocket < 0)

[math]\displaystyle{ u =v_{FR} }[/math] velocity of Fuel relative to the Rocket

[math]\displaystyle{ v = }[/math] velocity of rocket relative to the ground before ejecting fuel of mass [math]\displaystyle{ (-dm) }[/math]

[math]\displaystyle{ v+dv =v_{RG} }[/math] velocity of the rocket relative to the ground after ejecting fuel

[math]\displaystyle{ V_FG = }[/math] Velocity of the fuel with respect to the ground

Apply Conservation of Momentum

[math]\displaystyle{ mv = (-dm)V_{FG} + (m-(-dm))(v+dv) }[/math]

[math]\displaystyle{ dmV_{FG} = mdv + dm (v+dv) }[/math]

The velocity of the fuel with respect to the ground[math]\displaystyle{ (V_{FG}) }[/math] may be written as the vector sum of the rocket's velocity with respect to the ground [math]\displaystyle{ (V_{RG}) }[/math] and the velocity of the fuel with respect to the rocket [math]\displaystyle{ (V_{FR}) }[/math] :Galilean transormation

[math]\displaystyle{ \vec{V}_{FG} - \vec{V}_{FR} + \vec{V}_{RG} }[/math]

assuming 1-D motion and using the velocity variables defined above

[math]\displaystyle{ V_{FG} = -u + (v+dv) }[/math]

substituting

[math]\displaystyle{ dm \left (-u + (v+dv) \right ) = mdv + dm (v+dv) }[/math]

[math]\displaystyle{ -udm = mdv }[/math]

solving for the velocity

[math]\displaystyle{ \int_{v_0}^v = \int_{m_0}^m -u \frac{dm}{m} }[/math]

[math]\displaystyle{ v - v_0 = u \ln \left (\frac{m_0}{m} \right ) }[/math]

[math]\displaystyle{ v = v_0 + u \ln \left (\frac{m_0}{m} \right ) }[/math]

where [math]\displaystyle{ m_0 }[/math] is the initial Rocket mass

In terms of Thrust

Defining

[math]\displaystyle{ \mbox{THRUST} = - \dot{m} u }[/math]

Then

[math]\displaystyle{ m dot{v} = }[/math] THRUST = Force on rocket as a result of expelling fuel

Forest_UCM_MnAM#Rockets