Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Rocket equation

  1. Jul 15, 2008 #1
    recenty i read the rocket equation, derivation of, however i think i have a slight confusion with signs
    suppost initially a rocket has
    mass= [tex]M[/tex]
    velocity= [tex]\overrightarrow{v}[/tex]
    then at a time dt later,
    mass of rocket= [tex]M-dM[/tex]
    velocity of rocket= [tex]\overrightarrow {v} +d\overrightarrow {v} [/tex]
    mass of ejacted gas= [tex]dM[/tex]
    velocity of gas= [tex]\overrightarrow{u}[/tex]

    using conservation of momentum


    but [tex](\overrightarrow{u}-\overrightarrow{v})[/tex]=velocity of gas relative to rocket
    let [tex](\overrightarrow{u}-\overrightarrow{v})=\overrightarrow{U}[/tex]which is a constant


    now [tex]-ln\frac{M}{M_0}=\frac{\overrightarrow{v}-\overrightarrow{v_0}}{\overrightarrow{U}}[/tex]

    the problem is , when taking the velocity in the direciton rocket is travelling
    [tex]-ln\frac{M}{M_0}>0[/tex]since [tex]\frac{M}{M_0}<1[/tex]
    [tex]\overrightarrow{v}-\overrightarrow{v_0}<0[/tex] which is impossibe as the rocket is accelerating???
    Last edited: Jul 15, 2008
  2. jcsd
  3. Jul 15, 2008 #2
    latex problem seems to be fixed now...
    Last edited: Jul 15, 2008
  4. Jul 16, 2008 #3
    i dont see how you get [tex]
    M- dM
    [tex] is is because of [tex]E=mc^2[tex]
    Last edited: Jul 16, 2008
  5. Jul 16, 2008 #4
    why won't the latex work
  6. Jul 17, 2008 #5
    I hope this isn't too off topic but

    i think you need [/tex] instead of [tex] at the end
  7. Jul 17, 2008 #6

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    Good up to this point.
    This step is not valid. There is no such thing as the multiplicative inverse of a vector. Better is to define a unit vector v-hat directed along the rocket's delta-v vector. From the correct equation, this delta-v vector is directly opposed to the relative exhaust velocity vector. Thus

    d\overrightarrow{v} &= dv \hat{v} \\
    \overrightarrow{U} &= U \hat{v} & (U &\equiv \overrightarrow{U}\cdot \hat v)\\
    &= -v_e \hat{v} & (v_e&\equiv -U)

    Note that ve is simply the magnitude of the relative velocity vector. With this, the vector differential equation becomes the scalar equation


    from which

    [tex]\int_{M_0}^{M}\frac {dM}{M} = \frac 1{v_e}\int_{v_0}^v dv[/tex]


    [tex]\ln\frac{M}{M_0} = \frac{\Delta v}{v_e}[/tex]

    You can use a vector formulation, but you can't divide by a vector like you did.
  8. Jul 17, 2008 #7
    however is the equation
    [tex]v=v_0+\overrightarrow{U}ln\frac{M}{M_0}[/tex] not right?using the notion in the first post
    in the step with the integrals, to prevent multiplicative of inverse of vector, simply put [tex]\overrightarrow{U}[/tex]
    on the same side of the equation as [tex]\frac{dM}{M}[/tex]
    i have been told that its the problem associated with dM such that the mass of rocket after dt is M+dM not M-dM
    i dont get why minus can be used in scalar, but cannot be used in vercot derivation?
  9. Jul 17, 2008 #8

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    Your error occurs much earlier than I stated earlier.

    Here your dM is the quantity of mass ejected by the spacecraft. With this definition, a positive dM means the spacecraft loses mass. Things would have worked properly if you had used dM as positive meaning the spacecraft gains mass. Then the conservation of momentum equation becomes

  10. Nov 27, 2010 #9
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook