# What is the burn rate of the squares

## Conservation of momentum and shocks

#### Derivation of the equation of motion

Although a rocket ejects its propellant continuously, to derive the formula we consider a rocket that generates small amounts of fuel \ (\ Delta m \) in small periods of time \ (\ Delta t \)^{1} ejects; we will later justify our procedure more precisely, but it leads to the exact result.

The process of such a portion-wise ejection of fuel should be described from a resting observer and is in **Fig. 2** shown. At the time \ (t \), the observer sees the rocket with the mass \ (m \) flying upwards at the speed \ (v \) (we calculate upward speeds here as positive). In the following time span \ (\ Delta t \) the rocket hits the small amount \ (\ Delta m \)^{1} Propellant against its direction of movement, which reduces the mass of the rocket, but increases the speed of the rocket. At the end of this period of time, ie at the time \ (t + \ Delta t \), the observer sees the rocket with the mass \ (m - \ Delta m \) flying upwards at the speed \ (v + \ Delta v \) , but at the same time also fly the fuel with the mass \ (\ Delta m \) at the speed \ (u \) downwards (we calculate this speed as negative).

For the momentum change \ (\ Delta p \) caused by the fuel emission, \ [\ begin {eqnarray} \ Delta p & = & p (t + \ Delta t) - p (t) \ & = & \ left [ {\ left ({m - \ Delta m} \ right) \ cdot \ left ({v + \ Delta v} \ right) + \ Delta m \ cdot u} \ right] - m \ cdot v \ & = & \ left [m \ cdot v + m \ cdot \ Delta v - \ Delta m \ cdot v - \ Delta m \ cdot \ Delta v + \ Delta m \ cdot u \ right] - m \ cdot v \ & = & m \ cdot v + m \ cdot \ Delta v - \ Delta m \ cdot v - \ Delta m \ cdot \ Delta v + \ Delta m \ cdot u - m \ cdot v \ & = & m \ cdot \ Delta v - \ Delta m \ cdot v - \ Delta m \ cdot \ Delta v + \ Delta m \ cdot u \ & = & m \ cdot \ Delta v - \ Delta m \ cdot \ underbrace {\ left ({v + \ Delta v - u} \ right)} _ {{=: v _ {{\ rm {rel}}}}} \ quad (1) \ end {eqnarray} \] The size \ (v _ {\ rm {rel}} = {v + \ Delta v - u} \) is called **relative exit velocity** or **Outflow velocity**; it describes the speed at which the rocket ejects the fuel.

We now divide the above \ (\ Delta p \) by \ (\ Delta t \) and get \ [\ frac {{\ Delta p}} {{\ Delta t}} = \ frac {{m \ cdot \ Delta v - \ Delta m \ cdot {v _ {{\ rm {rel}}}}}} {{\ Delta t}} = m \ cdot \ frac {{\ Delta v}} {{\ Delta t}} - \ frac {{\ Delta m}} {{\ Delta t}} \ cdot {v _ {{\ rm {rel}}}} \] If we let \ (\ Delta t \) become smaller and smaller (and thus move away from portionwise Ejecting the fuel to the continuous ejection via), the difference quotients \ (\ frac {{\ Delta v}} {{\ Delta t}} \) and \ (\ frac {{\ Delta m}} { {\ Delta t}} \) by the differential quotients \ (\ frac {{dv}} {{dt}} \) and \ (\ frac {{dm}} {{dt}} \). We get \ [\ frac {{dp}} {{dt}} = m \ cdot \ frac {{dv}} {{dt}} - \ underbrace {\ frac {{dm}} {{dt}}} _ {=: \ mu} \ cdot {v _ {{\ rm {rel}}}} \] The size \ (\ mu = \ frac {{dm}} {{dt}} \)^{1} is referred to as **Mass flow** or **Throughput**; it describes how much fuel mass per unit of time is ejected by the rocket.

If an external force \ (F _ {\ rm {A}} \) acts on the rocket, such as gravitational force or air resistance, according to the general (and classical) formulation of NEWTON's 2nd axiom, \ (F _ {\ rm {A}} = \ frac {{dp}} {{dt}} \). This gives us \ [{F _ {\ rm {A}}} = \ frac {{dp}} {{dt}} \ Leftrightarrow {F _ {\ rm {A}}} = m \ cdot \ frac {{dv} } {{dt}} - \ mu \ cdot {v _ {{\ rm {rel}}}} \ Leftrightarrow m \ cdot \ frac {{dv}} {{dt}} = {F _ {\ rm {A}} } + \ underbrace {\ mu \ cdot {v _ {{\ rm {rel}}}}} _ {=: {F _ {{\ rm {thrust}}}}} \ Leftrightarrow \ frac {{dv}} {{ dt}} = \ frac {{{F _ {\ rm {A}}}}} {m} + \ frac {\ mu} {m} \ cdot {v _ {{\ rm {rel}}}} \] The Size \ (F _ {{\ rm {thrust}}} = \ mu \ cdot {v _ {{\ rm {rel}}}} \) is called **Thrust**. With \ (\ frac {{dv}} {{dt}} = a \) we finally get \ [a = \ frac {{{F _ {\ rm {A}}}}} {m} + \ frac {\ mu} {m} \ cdot {v _ {{\ rm {rel}}}} \] This is the rocket's equation of motion. Note that the acceleration \ (a = a (t) \) is generally not constant, because the right-hand side of this equation is not constant: the fuel ejection definitely changes the mass \ (m = m (t) \) with time. But the outflow velocity \ ({v _ {{\ rm {rel}}}} \) and the mass flow \ (\ mu \) can also change over time. Finally, the external force \ (F _ {\ rm {A}} \), e.g. in the case of gravitational force, can change with altitude or air resistance with altitude and speed. If these possible changes are taken into account, the equation of motion reads \ [a (t) = \ frac {{{F _ {\ rm {A}}} (v; h; t)}} {m (t)} + \ frac { \ mu (t)} {m (t)} \ cdot {v _ {{\ rm {rel}}} (t)} \ quad (*) \]

To make statements about the **Burn rate** \ ({v _ {\ rm {E}}} = v ({t _ {\ rm {E}}}) \) and the achievable height \ ({h _ {\ rm {E}}} = h ({t_ { \ rm {E}}}) \) at the time \ ({t _ {\ rm {E}}} \) - the so-called **Burnout time** - To be able to do it, you have to integrate the rocket's equation of motion. This method is usually only learned in mathematics lessons in high school.

^{1} As the mass of the missile over time **decreases**, the quantities \ (\ Delta m \), \ (\ frac {\ Delta m} {\ Delta t} \) and \ (\ frac {dm} {dt} \) are strictly negative. The size \ (\ mu \) is also often defined in the literature by \ (\ mu = - \ frac {dm} {dt} \). But since the mass of the rocket after the fuel ejection would have to be designated with \ (m + \ Delta m \) and the ejected fuel with \ (- \ Delta m \) (which all looks kind of strange), we calculate the above-mentioned quantities as positive . The result of our observations is still completely correct.

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