# Rocket Equation

What formula can be used in order to find the final velocity of a rocket if the time travelled by the rocket, exhaust velocity, initial and final masses of the rocket and the initial acceleration of the rocket is known (gravity is assumed to be constant)?

I came across the Tsiolkovsky rocket equation --> $$V_f= V_e \ln \frac {m_0}{m_f}$$. But why doesn't it include the time travelled by the rocket? is there another formula which also includes the time? Since the time travelled is also known to me in the particular case and I felt like I should also include it in a formula to reach the final velocity value.

• Hi Taylan, and welcome to the site! It looks like the equation you found satisfies your condition - that is, if you know the exhaust velocity, initial and final masses, and initial acceleration, it looks like you can use the equation you found to determine the final velocity. Could you elaborate more on why you think it shouldn't work? Or, if you really mean to ask why that equation doesn't include the time (which would be a good question that stands on its own), could you edit to make it more clear that that's the focus of your question? – David Z Nov 25 '18 at 12:41
• Hi David, thank you! I just edited the question to try to make it more clear. – Taylan Nov 25 '18 at 12:48

The mass of the rocket at a specific time is $$m_0 - \alpha t$$, where $$\alpha$$ is the rate of mass loss, so you can substitute this into the rocket equation to get $$v(t) = v_e \ln \frac{m_0}{m_0 - \alpha t}$$. The final velocity can then be obtained by checking the rocket at the time when it's ran out of fuel; this happens at $$\alpha t = m_0 - m_f$$.
The rate of mass loss is also known because you know the initial acceleration. From Newton's law, $$m_0 a = \frac{dp}{dt}$$, and in this case the momentum change comes from ejecting fuel, so $$a = \frac{\frac{dm}{dt} v_e}{m_0}$$, so $$\alpha = \frac{m_0 a}{v_e}$$.
Actually this equation comes from force equation of a variable mass system. It assumes that mass leaks at constant rate with a certain and a constant velocity relative to rocket which provides the thrust. Now you must certainly know the rate at which mass leaks. And of course you know the initial mass as $$(M=M_0-rt)$$ where $$r$$ is the rate and $$t$$ is time
Suppose the rate is not constant, then neglect gravity and apply conservation of momentum in vertical direction at a later time $$t$$ and at initial time. Find the equation of motion by solving the equation.