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Math Help - Rocket problem

  1. #1
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    Rocket problem

    Problem

    Consider a rocket with mass m, ejecting fuel at rate dm/dt with velocity u.
    The rocket will move forward with the velocity v, and the speed at which it will be moving forward must be such that the total momentum is conserved.
    This gives us the fundamental rocket equation

    m dv = u dm

    If the rocket starts from rest Vo=0 with initial mass Mi, then in order to find the speed that the rocket acquires, we need to integrate the left hand side of the equation from Vo to the final speed V and the right hand side of the equation from the initial mass Mi to the final mass Mf,

    v= u intgrl (dm/m).

    Evaluate the integral to find v. Also, find Mf (v)


    Attempted Solution:

    Is the integral of dm/m ln(Mf) - ln(Mi)? I.e. the natural logarith of the final mass divided by the initial mass?

    When they say find Mf(v), what does that mean?
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  2. #2
    MHF Contributor
    Joined
    Sep 2008
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    West Malaysia
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    Quote Originally Posted by brady951 View Post
    Problem

    Consider a rocket with mass m, ejecting fuel at rate dm/dt with velocity u.
    The rocket will move forward with the velocity v, and the speed at which it will be moving forward must be such that the total momentum is conserved.
    This gives us the fundamental rocket equation

    m dv = u dm

    If the rocket starts from rest Vo=0 with initial mass Mi, then in order to find the speed that the rocket acquires, we need to integrate the left hand side of the equation from Vo to the final speed V and the right hand side of the equation from the initial mass Mi to the final mass Mf,

    v= u intgrl (dm/m).

    Evaluate the integral to find v. Also, find Mf (v)


    Attempted Solution:

    Is the integral of dm/m ln(Mf) - ln(Mi)? I.e. the natural logarith of the final mass divided by the initial mass?

    When they say find Mf(v), what does that mean?
    hi this is my take

    From newton's 2nd law , F=dp/dt=d(mu)/dt

    Since momentum is conserved , F=0

    m(du/dt)+u(dm/dt)=0

    m du= - u dm

    \frac{du}{u}=-\frac{dm}{m}

    \int^{v}_{v_o}\frac{1}{u} du=-\int^{m_f}_{m_i} \frac{1}{m} dm

    \ln (\frac{v}{v_o})=-\ln (\frac{m_f}{m_i})

    \frac{v}{v_o}=\frac{m_i}{m_f}

    so move this around to get v and mf(v) respectively ,
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