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Variable mass: rocket equation
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Velocity change depends on exhaust speed and mass ratio.
Δv = v_e ln(m₀/m_f)Δv = vₑ ln(m₀/m_f): the speed gain grows only logarithmically with mass ratio, so very large fuel fractions are needed.
vₑ = 3 km/s, ratio 20: Δv = 9 km/s.
Higher exhaust speed matters more than adding fuel.
In more detail
The rocket equation Δv = v_e ln(m0/mf) gives the change in speed of a rocket from its exhaust speed and its initial and final masses. It follows from conservation of momentum with variable mass.
Real-world uses
- Mission planners use the rocket equation to size fuel loads for satellites.
- It explains why most of a rocket's mass is fuel.
- Space agencies use the rocket equation to plan how much propellant a mission needs to change its speed.
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Frequently asked questions
Why does the mass ratio matter so much?
Because the logarithm grows slowly, so a large mass ratio is needed for a large speed change.
Does the rocket need air to work?
No. It pushes against its own exhaust, so it works in vacuum.
Does a rocket need air to push against?
No. A rocket pushes against its own exhaust, so it works in the vacuum of space.
Related topics in Momentum & Collisions
Velocity of centre of mass · Centre-of-mass frame · Explosions and fragmentation · Thrust · Multi-stage rockets · Jet and conveyor problems