Rocket Dynamics
R. Wordsworth
September 20, 2016
1
Free space rocket equation
Consider a rocket with velocity v and total mass m that fires in deep space. At time t1 the
momentum of the rocket
p1 = mv.
(1)
The rocket fires propellant at velocity ve relative to the rocket. The propellant has mass −δm. At
time t2 the total momentum of the system (rocket + propellant) is
p2 = (m + δm)(v + δv) − δm(ve + v)
(2)
Note that the propellant velocity in the rest frame is ve + v, not ve . Note also the δm sign
convention: unless δm is negative, we are adding mass to the rocket.
Equating p1 and p2 we get
mv = mv + mδv + δmv − δmve − δmv + O[δ 2 ]
(3)
Here O[δ 2 ] just means ‘something of order δv or δm squared’. Hence
0 = mδv − δmve + O[δ 2 ]
(4)
dv = +ve d ln m
(5)
and taking limit δ → 0 we get
Solve this to get
Z
m
d ln m = −ve [ln m0 − ln m]
v − v0 = ∆v = +ve
(6)
m0
hence
∆v = −ve ln
hm i
0
.
(7)
m
So we get a change in velocity ∆v that is proportional to ve but in the opposite direction. This
makes sense!
Note if we define a mass flow rate of the rocket
b=−
then
m
dm
dt
dv
dm
= +ve
= −bve .
dt
dt
1
(8)
(9)
F = ma, so this is a force – the rocket’s thrust.
Also we can define specific impulse: Isp = ve /g0 , where g0 is (Earth’s) surface gravity. So the
scalarized version of (7) becomes
hm i
0
∆v = g0 Isp ln
.
(10)
m
Can also think of it as
Isp =
bve ∆t
ve
impulse
F ∆t
=
= .
=
propellant weight at Earth surface
∆mg0
b∆tg0
g0
(11)
Its usual units are seconds.
Remember thrust magnitude is bve , which explains why ion engines have low thrust despite
high Isp (b is very low).
2
Rocket equation with external forces
Easiest to start with equation in force form (9) and simply add an external term
m
dv
dm
= +ve
+ Fext .
dt
dt
(12)
Integrate to get
∆v = −ve ln
hm i
0
m
Z
+
0
t
Fext 0
dt .
m
Delta-v now also depends on (integrated) impulse per unit mass from external force.
Simplest case is vertical ascent in gravity field, Fext = mg. Then
Z t
hm i
0
∆v = −ve ln
+g
dt0 .
m
0
Scalarizing we get
hm i
0
− gt.
m
Last term is gravity loss or gravity drag. It decreases the delta-v, which makes sense.
∆v = +ve ln
2
(13)
(14)
(15)