A rocket in empty space has nothing to push against, yet it speeds up. It does it by throwing mass backward, a steady stream of Newton's Second Law for a System in action. Adding up the tiny kicks takes the logarithm from Integrals Giving Exponentials and Logarithms. Build one in the Rocket Builder lab.
Why it works
Watch from a frame moving with the rocket at one instant. In a short time the rocket (mass $m$) throws out a little exhaust, of mass $|dm|$, backward at speed $v_e$ relative to itself. Momentum in that frame starts at zero and stays zero, so the rocket gains $dv$ with $$m\,dv = v_e\,|dm| = -v_e\,dm,$$ since the rocket's mass is decreasing ($dm < 0$). Separate and integrate from the starting mass $m_0$ to the final mass $m_f$: $$\int dv = -v_e\int_{m_0}^{m_f} \dfrac{dm}{m} = v_e\ln\dfrac{m_0}{m_f}.$$ The push on the rocket is the momentum given to the exhaust each second: $F_{\text{thrust}} = v_e\left|\dfrac{dm}{dt}\right|$.
Key idea
$$\Delta v = v_e\ln\dfrac{m_0}{m_f}, \qquad F_{\text{thrust}} = v_e\left|\dfrac{dm}{dt}\right|.$$ The speed gained depends only on the exhaust speed and the mass ratio, not on how fast the fuel burns. Because of the logarithm, each extra bit of $\Delta v$ needs an exponentially larger mass ratio, $m_0/m_f = e^{\Delta v/v_e}$.
Worked example: Speed gained
A rocket in deep space is 80% fuel, with exhaust speed $3000\ \text{m/s}$. What is its $\Delta v$ when all the fuel is burned?
- $m_0/m_f = 1/0.2 = 5$.
- $\Delta v = 3000\ln 5 = 3000(1.609) = 4830\ \text{m/s}$.
Answer: About $4830\ \text{m/s}$.
Worked example: How much fuel?
A spacecraft with $v_e = 3000\ \text{m/s}$ needs $\Delta v = 6000\ \text{m/s}$. What fraction of its starting mass must be fuel?
- $m_0/m_f = e^{6000/3000} = e^2 = 7.39$.
- Fuel fraction $= 1 - 1/7.39 = 0.865$, so 86.5% of the starting mass is fuel.
- Doubling the $\Delta v$ again would need $e^4 \approx 55$: hence multistage rockets, which drop empty tanks along the way.
Answer: About 86.5%.
Common mistake
Thinking a rocket moves by pushing against the air or the launch pad. It works best in a vacuum, where there is nothing to push on at all. Check with momentum: the system "rocket plus exhaust" has no outside force in deep space, so if the exhaust gains backward momentum, the rocket must gain the same forward momentum. The push comes from the exhaust gas, which the rocket shoves backward and which shoves the rocket forward.
Near the ground, gravity also acts, so the rocket must have thrust larger than its weight to lift off, and $F_{\text{thrust}} - mg = ma$ gives its acceleration. The acceleration grows as fuel burns, since $m$ shrinks while the thrust stays about the same.