Rocket Trajectories

By Jan-Erik RønningenNorwegian Rocket Technology[ contact@rocketconsult.no ][ www.rocketconsult.no ]

Version: 1.50 2008

Contents Different types of Rocket Trajectories Typical Suborbital Trajectory Guided Flight Trajectory Homing Flight Trajectory Circular Orbits Gravity Rocket Mass Ratio Ideal Rocket Equation Rocket Equation with Drag, Thrust and Gravity (2) Terminal Velocity Trajectory Equations for a Single-Stage Rocket with

Thrust, Drag and Gravity Forces (5) Trajectory Simulation (2)

Different types of Rocket Trajectories Free Ballistic Flight Trajectory – Gravity and Drag

“Parabolic” type of trajectories, controlled by drag, gravity and thrust Arrows, dart, stone, bullet, sounding rocket

Guided Flight Trajectory – Preprogrammed Trajectory Trajectory shaped by i.e. lift, thrust vectoring and modulation Launch vehicles that places payloads into circular orbits

Homing Flight Trajectory – Programmed Trajectory during Flight Trajectory is uncertain and situation dependent. Shaped by thrust vectoring,

ailerons, side-thrusters, thrust modulation etc. Missiles with sensors that can detect target and have the ability to calculate a

meeting point in time and space in order to hit the target

Circular Orbits – Gravity and Speed Circular orbits are trajectories with almost infinity “free-fall”, shaped by speed

change and gravity Satellites, space crafts, meteorites, planets etc.

Typical Suborbital Trajectory - Single-Stage Sounding Rocket

Flight Profile: Parabola (actually part of an ellipse)

Powered Phase: Rocket thrust with time

Coasting Phase: Free flight up to apogee controlled by gravity and drag mainly

Free Fall: From apogee to ground impact (or splash down)

CRV-7 Forces at Burnout

100,0

18,20,7

Thrust [%]

Drag [%]

Gravity [%]

Different types of Rocket Trajectories Free Ballistic Flight Trajectory – Gravity and Drag

“Parabolic” type of trajectories, controlled by drag, gravity and thrust Arrows, dart, stone, bullet, sounding rocket

Guided Flight Trajectory – Preprogrammed Trajectory Trajectory shaped by i.e. lift, thrust vectoring and modulation Launch vehicles that places payloads into circular orbits

Homing Flight Trajectory – Programmed Trajectory during Flight Trajectory is uncertain and situation dependent. Shaped by thrust vectoring,

ailerons, side-thrusters, thrust modulation etc. Missiles with sensors that can detect target and have the ability to calculate a

meeting point in time and space in order to hit the target

Circular Orbits – Gravity and Speed Circular orbits are trajectories with almost infinity “free-fall”, shaped by speed

change and gravity Satellites, space crafts, meteorites, planets etc.

Guided Trajectory

R0

h

Different types of Rocket Trajectories Free Ballistic Flight Trajectory – Gravity and Drag

“Parabolic” type of trajectories, controlled by drag, gravity and thrust Arrows, dart, stone, bullet, sounding rocket

Guided Flight Trajectory – Preprogrammed Trajectory Trajectory shaped by i.e. lift, thrust vectoring and modulation Launch vehicles that places payloads into circular orbits

Homing Flight Trajectory – Programmed Trajectory during Flight Trajectory is uncertain and situation dependent. Shaped by thrust vectoring,

ailerons, side-thrusters, thrust modulation etc. Missiles with sensors that can detect target and have the ability to calculate a

meeting point in time and space in order to hit the target

Circular Orbits – Gravity and Speed Circular orbits are trajectories with almost infinity “free-fall”, shaped by speed

change and gravity Satellites, space crafts, meteorites, planets etc.

Homing Flight Trajectory

X

Different types of Rocket Trajectories Free Ballistic Flight Trajectory – Gravity and Drag

“Parabolic” type of trajectories, controlled by drag, gravity and thrust Arrows, dart, stone, bullet, sounding rocket

Guided Flight Trajectory – Preprogrammed Trajectory Trajectory shaped by i.e. lift, thrust vectoring and modulation Launch vehicles that places payloads into circular orbits

Homing Flight Trajectory – Programmed Trajectory during Flight Trajectory is uncertain and situation dependent. Shaped by thrust vectoring,

ailerons, side-thrusters, thrust modulation etc. Missiles with sensors that can detect target and have the ability to calculate a

meeting point in time and space in order to hit the target

Circular Orbits – Gravity and Speed Circular orbits are trajectories with almost infinity “free-fall”, shaped by speed

change and gravity Satellites, space crafts, meteorites, planets etc.

Circular Orbits

R0

a = g

G

v

Gravity

)2(sin0000059.0)(sin0052884.01780490.9 2220 s

mg

2

0

00

hR

Rggh

Earth is not a perfect sphere g will therefore be a function of latitude :

g also decreases with the square of the distance, sofor high altitudes a corrected g has to be calculated :

R0=6371315m

For h<120km R0 is so large in comparison to h that :

hggh 60 10086.3 [m] < 1% error

Rocket Mass Ratio

Propellant mass (md)

Rocket motor mass (mm)

Rocket inert mass (mr)

Payload mass (mp)

dmr

pb

mmmMM

mM

M

mMf

Burnout Mass

Start Mass

Ideal Rocket Equation

dm-ve

M-dmv

)ln(ln

lnln

1

max

max

0

max

feb

e

be

m

M

e

v

e

e

MvM

mvv

Mmvv

dmm

vdv

m

dmvdv

dmvdvm

b

+

Rocket Equation with Drag, Thrust and Gravity (1)

Rocket Equation with Drag, Thrust and Gravity (2)

bb

e

bbb

be

bbbe

ttmMe

vv

e

tgM

mvv

tgtmD

M

mvv

sm

tgtmD

Mmvvv

tgtmD

mv

dtgdtmD

dmm

vdv

bbb

0max

0max

0

00max

000

0

ln

to,simplifies

equation launch the verticaland drag noWith

sinln

,/0With v

0sin0lnln

sinlnv

gives,n Integratio

sin

sides,both on dt ith multiply w Then we

max

0

sin

sin

get, and

onefirst theinto equations theseinserts now We

as, same theison accelerati that know also We

as, described

becan rocket a of thrust that thelearned have We

sin

:equation thisreuse First we

getcan rocket a speed max. theFind

0

0

0

gmD

dtdm

mv

dtdv

gmD

m

vdtdm

dtdv

dtdv

a

vdtdm

F

gmD

mF

a

e

e

e

Negative because mass is expelled in oppositedirection of movement.

Terminal Velocity

0 GDNetForces

When drag equals gravity the netforces on the rocket is zero:

smCA

Gv

GvCA

Dt

D

/2

5.0 2

Terminal Velocity is then:

Higher velocity for objects that areheavier, more streamline, flying in lower atmosphere density and that has smaller frontal area.

Trajectory Equations for a Single-Stage Rocket with Thrust, Drag and Gravity Forces (1)

mrCoast

pmrThrust

mmM

mmmM

2

mr = rocket massmm= motor massmp = propellant mass

g = const. = 9.81m/s2

= const. = 1.22 kg/m3

CD = const.

Trajectory Equations for a Single-Stage Rocket with Thrust, Drag and Gravity Forces (2)

gmTz

vgmTdvm

dt

vgmT

dvmdt

dt

dvmDGT

dt

dvmamF

NvD

CA

Thrust

Thrust

Thrust

Thrust

Thrust

D

2

2

2

2

)(

][

5.0Constant:

max

max

022

22

22

ln2

1

max

vz

vz

z

mt

vz

dvmt

vz

dvmdt

vz

dvmdt

Thrustb

v

Thrustb

Thrust

Thrust

Drag:

tb= ? & vmax = ?

Simplify:

Burnout time:

Integration:

tb = It / Tavg

Normally tb is known.

NOTE:

Trajectory Equations for a Single-Stage Rocket with Thrust, Drag and Gravity Forces (3)

Finding vmax:

btw

btw

btw

btw

btwbtw

b

e

egmT

e

ezv

ezev

vz

vze

vz

vztw

vz

vztw

m

zw

Thrust

tw

b

b

Thrust

1

1

1

1

)1()1(

ln

ln

2

max

max

max

max

max

max

max

max

Trajectory Equations for a Single-Stage Rocket with Thrust, Drag and Gravity Forces (4)

We now want to find the burnout altitude, hb:

2max

2

2

022

022

22

2

2

ln2

)ln(2

max

max

vz

zmh

vzm

h

vz

dvvmh

dvvz

vmdh

gmTz

dh

dvvmvgmT

dh

dvvmF

dt

dhv

dt

dh

dh

dvm

dt

dvmamF

Thrustb

vThrustb

v

Thrustb

Thrust

Thrust

ThrustThrust

Thrust

Trajectory Equations for a Single-Stage Rocket with Thrust, Drag and Gravity Forces (5)

We now want to find the ascent altitude, ha and H:

2

2max

2

22

0

22

2

2

2

ln2

max

Coast

CoastCoastCoast

Coastv

Coast

Coast

Coast

CoastCoast

CoastCoast

Coast

Coast

z

vzmh

vz

dvvmdh

vz

dvvmdh

vgmdvv

mdh

gmz

vgm

dvvmdh

H = hCoast + hb

Trajectory Simulation (1)

Inp

ut

valu

esR

esu

lts

”Launch” Software Download:

http://users.cybercity.dk/~dko7904/software.htm

Trajectory Simulation (2)

Burnout

Apogee