Rocket Equations

[Pages:7]Rocket Equations

mR = rocket mass in kg

mE = engine mass (including propellant) in kg

mP = propellant mass in kg a = acceleration m/s2 F = force in kg . m/s2 g = acceleration of gravity = 9.81 m/s2 A = rocket cross-sectional area in m2

cd = drag coefficient = 0.75 for average rocket = air density = 1.223 kg/m3

= motor burn time in seconds T = motor thrust in Newton i.e. in kg . m/s2 I = motor impulse in Newton . seconds

v = burnout velocity in m/s hB= altitude at burnout in m hC= coasting distance in m mB = mR + mE - mP 2 mC = mR + mE - mP

mB g

mC g

k

=

1 2

cd

A

k v2

=I T

boost mass in kg coast mass in kg boost gravity force in kg . m/s2 coast gravity force in kg . m/s2

air drag coefficient in kg/m

air resistance in kg . m/s2

burnout time in seconds

Boost Phase: Burnout Time,Velocity and Altitude

F = ma

= m dv dt

= T - mg - k v2

m

dv dt

=

T

-

mg

-

k

v2

dt = m dv T - mg - k v2

=

m dv

k

T

- mg k

-

k

v2

= m dv k q 2 - k v2

= m dv k q 2 - v2

q2 = T - mB g k

q = T - mB g k

v

= m dv k q 2 - v2 0

=

mB k

1 2q

ln

q q

+ -

v v

2 kq mB

=

ln

q q

+ -

v v

p

=

ln

q q

+ v - v

-

p

=

ln

q q

- v + v

e - p = q - v q + v

(q + v ) e- p = q - v

q e- p + v e- p = q - v v + v e- p = q - q e- p

( ) ( ) v 1 + e- p = q 1 - e- p

v

=

q

1 - 1+

e - p e - p

F = ma

= m dv dt

= m dv dh dh dt

= m v dv dh

= T - mg - k v2

m v dv = T - mg - k v2 dh

dh = m v dv k q2 - k v2

dh = m v dv k q2 - v2

v

h = m v dv k q2 - v2 0

2kq = p mB

burnout velocity dh = v dt q2 = T - mB g

k q = T - mB g

k

[ ( )] hB

=

mB k

1 2

ln

q2

- v2

v 0

[ ( ) ( )] hB

= mB ln q 2 2k

- ln q 2 - v2

hB

=

mB 2k

ln

q

2

q2 -

v2

hB

=

mB 2k

ln T

T - mB g - mB g - k v2

Coast Phase: Altitude and Time

dh = mC v dv - mC g - k v 2

dh

=

mC

k

-

v mC

dv g -

k

v2

k

dh = mC v dv k qC2 - v 2

0

h C

=

mC k

v

v dv qC2 - v 2

hC

=

m 2k

ln

qC2 - qC2

v2

hC

=

m 2k

ln mC

g + k v2 mC g

F = m (- a)

= m - dv dt

= T - mC g - k v2

dt = - mC dv - mC g - k v2

dt

=

mC

mC

dv g+

k

v2

0

t C

=

mC k

v

dv qa2 + v 2

tC

=

mC k

1 qa

arctan

v qa

tC =

mC kg

arctan

v

mC g k

burnout altitude

- mC k

g

=

qC2

coast altitude

qa2

=

mC g k

time from v to 0

qa =

mC g k

ROCKET EQUATIONS gravit accel g air density rho drag coef cd rocket body mr engine empty ee propellant mp rocket total mt engine init me propellant p% mass flow m? exhaust v vex diameter d c-s-area A drag factor k

q qc? qa p

9.8100 m/s? 1.2230 kg/m^3 0.7500

0.0288 kg 0.0131 kg 0.0031 kg 0.0450 kg 0.0162 kg 6.9333 % 0.0134 kg/s 800.0000 m/s 0.0250 m 0.00049 m? 0.00023 kg/m 213.4473 m/s -1824.93776 42.7193 m/s 2.2124 1/s

thrust

T

impulse

I

boost mass mb

coast mass mc

burnout time tau

velocity b vt

altitude b hb

altitude c hc

altitude t ht

coast time tc

max velocity vkm/h

max air drag Fd

check

burnout time tau

10.6829 N=kg*m/s? 2.4960 N*s=kg*m/s 0.04344 kg 0.04188 kg 0.23364 s 53.97011 m/s 6.37419 m 88.73626 m 95.11045 m 3.92459 s 194.29240 km/h 0.65574 N=kg*m/s?

0.23364486 s

Influence of Exhaust Velocity 250

200

150

m/s m

100

50

0 500

750

1000

1250

1500

1750

2000

2250

Exhaust Velocity in m/s

vt

hb

hc

ht

m/s m

100 90 80 70 60 50 40 30 20 10 0 0.0020

0.0040

Influence of Mass Flow

0.0060

0.0080

0.0100

Mass Flow in kg/s

0.0120

vt

hb

hc

ht

0.0140

Influence of Propellant Mass 350

300

250

200

m/s m

150

100

50

0 0.0000

0.0020

0.0040

0.0060 0.0080 mp in kg

vt

hb

hc

0.0100 ht

0.0120

0.0140

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