The Rocket Equation

• Defines the change in velocity of a rocket as it expels a propellant

• Derived from Newton’s Second and Third Laws

• Can be derived from Conservation of Momentum and Energy

vf – vi = vrel ln (mi/mf)

vf final velocity of the rocket in m/s

vi initial velocity of the rocket in m/s

vrel relative velocity between the exhaust and

the rocket in m/s

ln natural logarithm

mi initial mass of the rocket in kg

mf final mass of the rocket in kg

Deriving the Rocket EquationDeriving the Rocket Equation

Begin with F = dp/dt (Newton’s Second Law)Recall, p = m v (momentum)Do not assume m nor v are constant, so

F = d(m v)/dt = v dm/dt + m dv/dtIf we let the force go to zero (equilibrium),

-v dm/dt = m dv/dtReworking, dv = -v dm/mIntegrating, we find the rocket equation

vf – vi = vrel ln (mi/mf)