microacg wrote:After a few steps there is a need to integrate: dv / [ g/v - alpha*v ]. According to the text, "Integrating (test by substitution), the velocity is: (answer). How did they integrate that expression? I asked one person who suggested multiplying the top and bottom of the fraction by v and then using partial fractions, although that seems different than what the author did.
2) Page 24 features a derivation of the rocket motion equations. For the case of gravity: m(t) * dv/dt = v_naut * m_dot - m(t) * g. Using separation of variables, both sides of the resulting equation is integrated. The left side is simply dv integrated to yield v(t). The right side expression, culminating in dt, is: m_dot/(m_naut - m_dot * t) * v_naut - g. The book integrates the expression from 0 to t. My first thought was to simply take the indefinite integral. However, if you take the indefinite integral you seem to get a different answer for the integration (due, I think, to the u substitution I used). What is the difference between integrating the expression from 0 to t, and taking the indefinite integral?
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