## Physics of Rocketry: Distance Traveled

Calculate the firing times for the acceleration and deceleration leg of the mission described below and also the distance traveled for each leg. I also need to calculate the initial mass of the vehicle.

1) acceleration leg from zero velocity to a final velocity of 25,000 m/s.
2) 360 'cruise' non-acceleration leg (constant velocity) at 25,000 m/s; systems shut down.
3) deceleration leg (delta V=25,000 m/s)

This question is part of a bigger question of which I had to solve a couple things for. What's given:

ISP=4000 seconds
g=9.81 m/s^2
$$m_L$$=7000 kg

I have calculated the mass flow rate of the propellant to be .006371 kg/s and the mass of the engine to be 3.42485 kg. Here is my work thus far.

initial mass=mass of engine + mass of payload + mass of propellant
$$m_0=m_e+m_L+m_p$$

$$\frac{m_0-m_p}{m_0}=e^{\frac{-\Delta V}{ISPg_0}}}$$

So I solve the above equation for $$m_p$$ and put into the first equation to obtain the equation below solved for $$m_0$$.

$$m_0=\frac{m_e+m_L}{e^{\frac{-\Delta}{ISPg_0}}}$$

So plugging in the numbers where $$\Delta V$$=25000, I get the initial mass of the vehicle to be 13243.5 kg. This seems reasonable.

Now this is where I get confused, how do I go about solving for the propellant mass in each leg and the distance traveled. I believe that I have to do some sort of integration for the distance traveled due to the mass constantly decreasing. Is there an equation or integration set up for this already?

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