A rocket accelerates by ejecting mass with velocity relative to the rocket. In the ideal one-dimensional case, let

• $m$ be the rocket's instantaneous mass, including remaining propellant;
• $v$ be its velocity in an inertial frame;
• $u_e>0$ be the constant exhaust speed relative to the rocket, directed backward.

Take the positive direction along the rocket's motion. The exhaust relative velocity is therefore

$$u=-u_e,$$

while the rocket loses mass, so

$$dm<0.$$

With no external force, the variable-mass momentum balance gives

$$m\frac{dv}{dt}=u\frac{dm}{dt}=-u_e\frac{dm}{dt}.$$

Thus

$$dv=-u_e\frac{dm}{m}.$$

Integrating from initial mass $m_i$ and velocity $v_i$ to final mass $m_f$ and velocity $v_f$,

$$\int_{v_i}^{v_f}dv =-u_e\int_{m_i}^{m_f}\frac{dm}{m}.$$

Therefore

$$v_f-v_i =-u_e\ln\left(\frac{m_f}{m_i}\right),$$

or

$$\boxed{\Delta v=u_e\ln\left(\frac{m_i}{m_f}\right)}.$$

This is the ideal rocket equation.

Mass ratio matters logarithmically

Define the mass ratio

$$R=\frac{m_i}{m_f}>1.$$

Then

$$\Delta v=u_e\ln R.$$

Increasing exhaust speed improves $\Delta v$ directly, while increasing mass ratio gives diminishing returns because the dependence is logarithmic.

Example

A rocket has effective exhaust speed

$$u_e=2500,\mathrm{m/s},$$

initial mass

$$m_i=10{,}000,\mathrm{kg},$$

and final mass after the burn

$$m_f=4000,\mathrm{kg}.$$

Its ideal velocity change is

$$\Delta v =(2500)\ln\left(\frac{10{,}000}{4000}\right) =(2500)\ln(2.5) \approx2.29\times10^3,\mathrm{m/s}.$$

So the burn can change the rocket's speed by about

$$2.29,\mathrm{km/s}$$

under the ideal assumptions.

Thrust

The instantaneous thrust magnitude for constant exhaust speed is

$$\boxed{T=u_e\dot m_e},$$

where

$$\dot m_e=-\frac{dm}{dt}>0$$

is the propellant mass ejection rate. A larger mass flow produces larger thrust, but the total ideal $\Delta v$ for a completed burn depends on exhaust speed and mass ratio rather than directly on how quickly the propellant is expelled.

What the ideal equation omits

The derivation assumes constant exhaust speed and no external forces. Gravity, aerodynamic drag, pressure effects, steering losses, and changing exhaust performance can all alter the actual velocity change of a real launch vehicle.

The rocket equation is therefore not a separate conservation law. It is the integrated consequence of momentum conservation for a body whose mass decreases while exhaust leaves with a specified velocity relative to that body.