# How would one calculate velocity with respect to time in Case 2?

Case 1. : The rocket is weightless in space. It's fueled with propellant. It's wet mass is starting mass $$m_s$$ and dry mass final mass $$m_f$$. Suppose rocket's initial velociti is 0. The rocket ignites the engines which provide constant thrust $$T$$ during the flight for some time $$t_f$$ as for final time. We can now derive the following equation for mass of the rocket $$m$$ with respect to the time elapsed $$t$$ : $$m(t)=-t*\frac{(m_s-m_f)}{t_f}+m_s$$
Using Newton's seccond law of motion we can write the equation for acceleration with respect to time : $$a(t)=\frac{T}{-t*\frac{(m_s-m_f)}{t_f}+m_s}$$
If we now integrate acceleration with respect to $$t$$, we get velocity with respect to time : $$v(t)=-T\cdot\frac{t_2}{m_1-m_2}\cdot\ln\left(\left|m_1-\frac{m_1-m_2}{t_2}\cdot t\right|\right)+C$$, where $$C=v(0)$$
If we graph these equations we get a beatiful representation of rocket's mass, acceleation and velocity with respect to time. ->[1]: https://i.stack.imgur.com/eqZkq.png

Case 2. : This is my question -> What if the rocket starts from a launchpad and travells so high that acceleration due to gravity $$g$$ starts changing. This then affects rocket's acceleration which affects rocket's velocity but rockets velocity is dependend on acceleration due to gravity. Everything seems to be connected and there seems to be no direct equation we can derive to get the rocket's velocity. I'd really like to know the methods and thinking that goes into modelling this problem.

• Use this equation $m\left( t\right) a=T-m\left( t\right) \cdot g$ – Eli Jul 15 at 16:52
• Hey! I don't see how that would provide me with a velocity equation. You seem to be assuming g to be a constant, but I want to know how would one express velocity if both mass and g are changing. – ToTheSpace 2 Jul 15 at 17:21
• g changing ? Mass ist changing? – Eli Jul 15 at 17:27
• I am assuming the rocket is sitting on a launchpad so at t=0 g=9.81m/s^2 is acting on a rocket. As the rocket gains altitude, the mass of the rocket keeps decreasing due to rocket ejecting propellant and gravitational constant g starts decreasing due to increasing distance between rocket and the Earth thus acceleration of the rocket doesn't increase only due to decreasing mass but also due to decreasing gravitational force. How would we combine those two phenomena into one equation which would yield velocity with respect to time? – ToTheSpace 2 Jul 15 at 17:34
• You have to solve this differential equation $m\left( t\right) \cdot \ddot r=T-\dfrac {m\left( t\right) \cdot MG}{r^{2}}$ – Eli Jul 15 at 17:50