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How fast is fuel escaping a rocket for it to reach the escape velocity 11 km/s?

how fast is a rocket
habboin 01/02/2022 Rocket 853
$\begingroup$For a rocket that is not subject to external forces, conservation of momentum dictates that $$m(t)\,\dot v(t) + v_e(t)\,\dot m(t) = 0$$ where$m(t)$ is the mass of the rocket, including pr...

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For a rocket that is not subject to external forces, conservation of momentum dictates that $$m(t)\,\dot v(t) + v_e(t)\,\dot m(t) = 0$$ where

$m(t)$ is the mass of the rocket, including propellant, at time $t$,$v(t)$ is the rocket's velocity at time $t$, relative to some inertial observer,$\dot v(t)$ is the rocket's acceleration at time $t$,$v_e(t)$ is the velocity at which the rocket expels exhaust, relative to the rocket, and$\dot m(t)$ is the rate at which the rocket is losing mass.

Assuming a constant exhaust velocity, integrating this with respect to time results in the ideal rocket equation for a single stage of a rocket: $$\Delta v = v_e \ln\left(\frac{m_0}{m_f}\right)$$ where

$\Delta v$ is the rocket's change in velocity,$v_e$ is the effective velocity at which exhaust leaves the rocket,$m_0$ is the rocket's initial mass (payload, structure, and propellant),$m_f$ is the rocket's final mass (payload and structure), and$\ln(x)$ is the natural logarithm function.

This means that having a rocket's change in velocity exceed the rocket's exhaust velocity is eminently achievable. It requires that the rocket's initial mass be at least 63% propellant. Rockets that propel things into space typically have an initial mass that is about 90% propellant.

As a rule of thumb, trying to make a single stage of a rocket have a delta V that is more than three times exhaust velocity is pushing the rocket equation a bit too hard. That would require, at a minimum, a rocket whose initial mass is 95% propellant. A more realistic value is a rocket whose initial mass is in the neighborhood of 90% propellant. This results in a rocket that ideally has a delta V that is 2.3 times exhaust velocity.

This, by the way, is why the concept of a single stage to orbit rocket is appealing and yet seemingly just out of grasp. The delta V needed to put a payload into low Earth orbit ranges from over 9 km/s to a bit over 10 km/s, depending on the rocket. A rocket whose initial mass is 90% propellant and whose exhaust velocity is 4 km/s (both of which are feasible) can ideally achieve a delta V of 9.2 km/s. This is just in range of what is feasible. The problem is the word "ideally". That a single stage to orbit rocket is just on the cusp of what is feasible means that organizations that want to put things into orbit or beyond inevitably take on the significant added complexities associated with multistage rockets.