Results 1 to 6 of 6

Math Help - Problem with integating for rocket boost phase, can anyone help?

  1. #1
    Newbie
    Joined
    May 2010
    Posts
    3

    Problem with integating for rocket boost phase, can anyone help?

    I need to integrate this function once with respect to t to find an expression for h(t):
    v(t) = -k ln(m-bt) - gt + k ln(m)

    Can anyone show me how to do this? How would I resolve the constant?

    I then need to show that, after substituting t into h(t) I get the expression:

    h(t) = km/b [(1 - X) ln(1 - X) + X(1 - (g/2kb)Xm0)]

    Sorry I wasn't sure how to put that into LaTeX, I hope it's clear enough in that form.

    I should mention also that k, b, m and g are constants. This is all to find an expression for h(t) when the rocket reaches the end of its boost phase. Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Haven's Avatar
    Joined
    Jul 2009
    Posts
    197
    Thanks
    8
    You need an initial condition to resolve the constant.

    In this case I would assume that h(0) = 0.
    Substituting the constant in you can solve for zero.

    The LaTeX commands you want are \ln{x} for \ln{x}
    and \frac{1}{2} for \frac{1}{2}
    and for subscripts a_n for a_n
    to adjust bracket size use the following commands:
    \left[ and \right] so  [\frac{1}{2}] becomes \left[\frac{1}{2}\right]
    So your formulas become:
    v(t) = -k\ln(m-bt) - gt + k\ln(m)
    h(t) = \frac{km}{b} \left[(1 - X)\ln(1 - X) + X(1 - \left(\frac{g}{2kb}\right)Xm_0)\right]
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2010
    Posts
    3
    Ok, thanks for your reply. However would you be able to show the steps to get the expression of h(t) from the v(t) expression?

    Also how would I find t*?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,973
    Thanks
    1121
    \int \ln(x) dx= x \ln(x)- x so to integrate \int -k \ln(m-bt) - gt + k \ln(m)= -k\int \ln(m- bt) dt- g\int t dt+ k\ln(m)\int dt, let u= m-bt in the first integral. Then du= -bdt so dt= -(1/b)dt and the integeral becomes \frac{k}{b}\int \ln(u)du= \frac{k}{b}(u \ln(u)- u)= \frac{k}{b}((m-bt)\ln(m- bt)- (m- bt)).

    The entire integral is
    h= \frac{k}{b}((m-bt)\ln(m- bt)- (m- bt))-\frac{g}{2}t^2+ k\ln(m)t+ C

    As Haven said, set h(0) equal to the initial height to determine C.

    You ask how to find t* but don't define t*. If you mean "at the end of the boost phase" then you will need some other condition to determine what t* itself is before putting it into that equation.
    Last edited by HallsofIvy; May 28th 2010 at 02:35 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    May 2010
    Posts
    3
    Thanks for your help! Yeah I did mean at the end of the boost stage. When the fuel runs out the equation is:

    m(t*) = -bt + m = m - Xm

    where m is the initial mass of the rocket and and the initial mass of fuel is Xm, where X is an unspecified fraction between 1 and 0.

    How would I find an expression for t? Also how would I go about putting it into the expression for h(t) and v(t)?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,973
    Thanks
    1121
    Quote Originally Posted by Kiche View Post
    Thanks for your help! Yeah I did mean at the end of the boost stage. When the fuel runs out the equation is:

    m(t*) = -bt + m = m - Xm

    where m is the initial mass of the rocket and and the initial mass of fuel is Xm, where X is an unspecified fraction between 1 and 0.

    How would I find an expression for t? Also how would I go about putting it into the expression for h(t) and v(t)?
    When you are doing problems involving integrals of logarithms, you are expected to know how to solve things like -bt*+ m= m-Xm!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. 3 second burn rocket problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 9th 2011, 07:18 PM
  2. Phase Space/ State Space/ Phase Portrait
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: October 12th 2010, 01:40 PM
  3. Rocket linear equation problem
    Posted in the Algebra Forum
    Replies: 1
    Last Post: September 17th 2010, 04:15 PM
  4. Phase plane problem
    Posted in the Differential Equations Forum
    Replies: 7
    Last Post: June 10th 2010, 02:55 AM
  5. Rocket problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 23rd 2010, 10:05 PM

Search Tags


/mathhelpforum @mathhelpforum