Rockets are variable-mass systems. As propellant is expelled backward, the rocket mass decreases and the remaining
rocket gains forward velocity. The ideal result is the Tsiolkovsky rocket equation.
Ideal rocket equation
For an ideal rocket in vacuum with no gravity loss, no drag, and constant exhaust velocity:
\[
\Delta v=v_e\ln\left(\frac{m_0}{m_f}\right)
\]
where \(v_e\) is exhaust velocity, \(m_0\) is initial mass, and \(m_f\) is final mass after propellant burn.
Mass ratio
The mass ratio is:
\[
R=\frac{m_0}{m_f}
\]
The rocket equation can also be written:
\[
\Delta v=v_e\ln R
\]
This logarithm is why very large delta-v requires extremely large mass ratio or very high exhaust velocity.
Exhaust velocity and specific impulse
Exhaust velocity is related to specific impulse by:
\[
v_e=I_{sp}g_0
\]
where \(g_0\approx9.80665\ \mathrm{m\,s^{-2}}\). Higher \(I_{sp}\) means more delta-v for the same mass ratio.
Thrust and mass flow
For constant thrust and exhaust velocity:
\[
T=\dot m v_e
\]
so the propellant mass-flow rate is:
\[
\dot m=\frac{T}{v_e}
\]
If the propellant mass is \(m_p\), the burn time is:
\[
t_b=\frac{m_p}{\dot m}
\]
Gravity loss
In a vertical launch, thrust must fight gravity during the burn. A simple estimate is:
\[
\Delta v_g\approx g\,t_b
\]
More generally, if the flight-path angle above the horizontal is \(\gamma\):
\[
\Delta v_g\approx g\sin(\gamma)t_b
\]
A real launch trajectory changes angle over time, so this is an approximation.
Drag loss
Atmospheric drag is approximated by:
\[
D=\frac12\rho C_D A v^2
\]
The acceleration lost to drag is:
\[
a_D=\frac{D}{m}
\]
Since \(v\), \(m\), and air density can all change during flight, drag loss is best estimated by numerical integration.
Multistage rockets
Multistage rockets improve performance by dropping empty dry mass after a stage has burned. For each stage:
\[
\Delta v_i=v_{e,i}\ln\left(\frac{m_{0,i}}{m_{f,i}}\right)
\]
Total ideal delta-v is the sum of stage contributions:
\[
\Delta v_{\mathrm{total}}=\sum_i \Delta v_i
\]
In a staged rocket, the start mass of each stage includes all upper stages and payload still above it.
Worked example
Suppose \(v_e=3000\ \mathrm{m\,s^{-1}}\), \(m_0=5000\ \mathrm{kg}\), and \(m_f=2000\ \mathrm{kg}\).
Step 1: mass ratio.
\[
R=\frac{m_0}{m_f}=\frac{5000}{2000}=2.5
\]
Step 2: ideal delta-v.
\[
\begin{aligned}
\Delta v
&=
v_e\ln R\\
&=
3000\ln(2.5)\\
&\approx
2749\ \mathrm{m\,s^{-1}}
\end{aligned}
\]
So the ideal delta-v is about \(2.75\ \mathrm{km\,s^{-1}}\). This is the pure vacuum value, before gravity and drag losses.
Formula summary
| Concept |
Formula |
Meaning |
| Ideal rocket equation |
\(\Delta v=v_e\ln(m_0/m_f)\) |
Ideal velocity gain from propellant burn. |
| Mass ratio |
\(R=m_0/m_f\) |
Initial mass divided by final mass. |
| Specific impulse relation |
\(v_e=I_{sp}g_0\) |
Connects exhaust velocity and engine efficiency. |
| Thrust relation |
\(T=\dot m v_e\) |
Thrust from exhaust mass flow. |
| Mass flow |
\(\dot m=T/v_e\) |
Propellant consumed per second. |
| Burn time |
\(t_b=m_p/\dot m\) |
How long the stage burns. |
| Gravity loss estimate |
\(\Delta v_g\approx g\sin(\gamma)t_b\) |
Velocity lost while fighting gravity. |
| Drag force |
\(D=\frac12\rho C_DAv^2\) |
Atmospheric resistance. |
| Multistage total |
\(\Delta v_{\mathrm{total}}=\sum_i v_{e,i}\ln(m_{0,i}/m_{f,i})\) |
Total ideal delta-v from all stages. |
The rocket equation is logarithmic: adding more propellant gives diminishing returns unless dry mass is staged away or exhaust velocity improves.
