How to calculate rocket acceleration?

Question

What are the equations used to calculate rocket thrust, acceleration, and weight? Obviously this depends on the type of fuel and engine and a lot of other factors, but I'm interested in a simplified ideal case.

For example, a solid rocket booster would provide a constant force thrust, but it would get lighter as it burns up its fuel, so it's thrust-to-weight-ratio would change mid flight, and acceleration would increase.

Answer 1

This is for idealized case (see below), but Tsiolkovsky rocket equation should have you covered:

$$\Delta v = v_\text{e} \ln \frac {m_0} {m_1}$$

where:

• $m_0$ is the initial total mass, including propellant,
• $m_1$ is the final total mass,
• $v_\text{e}$ is the effective exhaust velocity,
• $\Delta v$ is delta-v - the maximum change of velocity of the vehicle (with no external forces acting),
• $\ln$ refers to the natural logarithm function.

There's many online rocket equation calculators to simplify this for you, for example this Delta-V Calculator, and you can find many more on Atomic Rockets Online Calculators page.

And average acceleration over a period of time is then:

$$\boldsymbol{\bar{a}} = \frac{\Delta \mathbf{v}}{\Delta t}$$

Effective exhaust velocity is mostly given in Specific Impulse in:

$$I_{\rm sp}=\frac{v_{\rm e}}{g_{\rm 0}}$$

where:

• $Isp$ is the specific impulse measured in seconds
• $v_{\rm e}$ is the average exhaust speed along the axis of the engine (in ft/s or m/s)
• $g_0$ is the acceleration at the Earth's surface (in ft/s2 or m/s2).

And thrust is:

$$T=v_{\rm e}\frac{\Delta m}{\Delta t}$$

where:

• $T$ is the thrust generated (force),
• $\frac{\Delta m}{\Delta t}$ is the rate of change of mass with respect to time (mass flow rate of exhaust), and
• $v_{\rm e}$ is the speed of the exhaust gases measured.

You can derive your own equations out of these ones, based on the data that you have available.

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Do note that, as mentioned previously, this is idealized case and in reality none of your input data will remain constant during entire flight, not merely T/W ratio. If, for example, you have a thrust profile as a function of time available, these same equations apply, but you'll have to recalculate for each change in thrust, which will give you other graphs, e.g. mentioned T/W ratio as a function of time, and so on. And there's variables that will change with altitude, velocity (e.g. drag coefficient), attitude and even time since some specific event (e.g. propellants boil-off rate, grain geometry of solid motors as mentioned by Adam in the comments, fuel-to-oxidizer mixture ratio, thermal expansion, ad nauseum,...). The list of variables is in reality infinite, and it's due to these that rocket science isn't as easy as ABC, as these few listed basic equations might suggest. The list of all equations that you might need is in reality unmanageably long.