# How do I calculate a rockets burn time from required velocity?

Say my rocket could produce F newtons of thrust while consuming M1 kilograms of fuel per second. It's mass at start is M0, and it keeps burning until it reaches V velocity.

How would I find out the necessary burn time for the rocket to reach V velocity?

Alternate Wars gives this formula for computing the length of a rocket burn:

$$\Delta T = \frac {M_L E_V} {F} (1 - e ^ {-\frac {\Delta V } {E_V}})$$

Where:

$\Delta T$: Length of burn in seconds

$M_L$: Total mass of the rocket at the beginning of the burn (often written $m_0$)

$E_V$ = Exhaust Velocity in meters/second (often written as $v_e$).

$F$: Thrust of the rocket in Newtons.

$\Delta V$ = Delta-V of burn in meters/second.

Your $M_0$ is this equation's $M_L$. Exhaust velocity $E_V$ is equivalent to thrust divided by mass flow rate (that's your $F$ and $M_1$).

Exhaust velocity is one of two standard forms for representing mass-specific impulse. More often you'll see specific impulse called $I_{sp}$ and measured in seconds (but really meaning pounds-of-force-seconds-per-pound-of-mass); $I_{sp}$ in seconds times gravity at Earth's surface ($\approx 9.81 \frac {m} {s^2}$) yields exhaust velocity.

If you know the$\Delta v$, you can estimate the fuel mass needed via the rocket equation $\Delta v=v_{e}log(\frac{m}{m_f})$, where your propellant mass would be $m-m_f$. If you also know the mass consumption rate $\dot{m}$, then you could divide the two together.

The key is knowing how to estimate the effective exhaust velocity. If you know the thrust $T$, you can estimate it by $v_e=\frac{T}{\dot{m}}$, where $\dot{m}$ is the mass flow rate.

• @RussellBorogove: You’re right. I should have written the mass flow rate. I updated it. – Paul May 21 '18 at 23:50
• Sorry to be that guy, but it's just $\frac T {\dot m}$ in metric units. $g_0$ is only needed if your mass and force units are the same (e.g. pounds-force vs pounds-mass). – Russell Borogove May 21 '18 at 23:51
• @RussellBorogove: it’s fine to be “that guy”. I learn best when i get feedback:) – Paul May 22 '18 at 0:01