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.
I don't have the book with me, but I did check the integral on Wolfram Alpha. It says to multiply by v/v as your friend suggested and then simply use a substitution u = g-alpha*v^2 so du = -2*alpha*vdv (which is the top, with some constants) and the bottom is simply u, so you then just integrate du/u.
By the way, Wolfram Alpha is my best friend for integrations because I'm not a real physicist (and soon, not even a real astronomer!)
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?
When I do the separation of variables to get dv on the left hand side, I get:
[ v0 *mdot/m(t) - g ] * dt
on the right hand side, instead of what you wrote. (v0 = v_naught). I can then write the right hand integral as the difference of two integrals (I think?). The second integral is easy (integrate g*dt = g*t). For the first one, I write mdot = dm/dt and then the dt's cancel out, so you just have to integrate v0*dm/m, which is v0*ln[m(t)].
Without the book, I'm not sure what m and v_naught represent (i.e. mass of rocket + fuel, or just rocket, or just fuel, etc. and is v_naught the velocity of the rocket or fuel?) But that's the integrated value I get if I just integrate the equation you give me. I didn't check this very carefully though so I could have made a mistake too!