Question

A rocket ascends from rest in a uniform gravitational field by ejecting exhaust with constant speed u. Assume that the rate at which mass is expelled is given by dm/dt=γm, where m is the instantaneous mass of the rocket and γ (gamma) is a constant, and that the rocket is retarded by air resistance with a force bv, where b is a constant. Find the v(t). While this is technically a Physics problem, the part I'm stuck on at this point is entirely math (differential equation solving). Look in the comments to see the differential equation I got. Does anyone know how to solve for v?

Answer 1

It can be shown that, given an initial mass m_0:\[m=m_0 e^{\gamma t}\]Therefore, Newton's second law tells us \[ma = \dfrac{dp}{dt}-mg-bv\]\[m\dot v = u\dfrac{dm}{dt}-mg-bv\]\[m\dot v = u\gamma m-mg-bv\]\[\dot v + \dfrac{b}{m}v+(g-u\gamma)=0\]\[\dot v + bm_0e^{-\gamma t}v+(g-u\gamma)=0\]Please help me find v(t) where \[\dot v = \dfrac{dv}{dt}\]D:

Answer 2

In short, solve for v(t) where \[\dfrac{dv}{dt}+bm_0e^{-\gamma t}v+(g-u\gamma)\]and b, m_0, γ, g, and u are constants (γ is probably negative, with the other ones positive).