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Data for individual stages.

*derived from T/R

**Derived from m⁰_fuel/t_final

***First of two separate burns; second burn 312 seconds.

Total fuel for third stage = 0.103x10⁶ kg with total burn time of 477 seconds giving a net fuel burn rate of 216 kg/s

Stage 1 Saturn V. 

Although it was launched from the Earth’s surface,

 gravity is neglected in these calculations.

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Stage 1— On the launch pad fully laden with fuel and payload.

          

m⁰_fuel  = 2.04x10⁶ kg  initial amount of fuel

m_body = 2.77x10⁶ kg (everything on launch pad) - m⁰_fuel  =  0.73x10⁶ kg

m_initial =  2.77x10⁶ kg  = m_body +m⁰_fuel

v_exhaust = 2456 m/s

mass of the frame to jettison = 0.136x10⁶ kg

since all fuel is consumed, m_final = m_body

R = 13600 kg/s

t_final = 150 seconds = the duration of the burn

 

*linear displacement

**4.4 second coast period between first and second stages;                            displacement given by x_coast = t_coast  v_final

Stage 1 Notes

v_final = 3275 m/s = 7300 miles per hour

a(t=0) = 12.1 m/s² = 1.2 g’s

a(t_final) = 4.7 g’s   on track here; NASA says the rocket could generate 4.5 g’s at the end of Stage 1

x(t_final) = 120 miles

Stage 1 Results and Graphs — no gravity

The results of this calculation give a first stage final distance of 120 miles with a speed of 7300 miles/hour, whereas NASA says that the first stage of the Saturn V ends at an altitude of 38 miles with a speed of 5330 miles/hour. 

Without gravity the rocket would be travelling at more than seven thousand miles per hour — after only the first stage! — and after only two and one-half minutes!   I would like to know what the speed after the third stage would be.

Stage 1 Saturn V.

Taking into account the component of gravity along the direction of motion

using the empirical method given on page 4.

Tau = 256 seconds is selected to match the velocity curve to NASA velocity data at end of Stage 1.

*linear displacement

**4.4 second coast period between first and second stages;                            displacement given by x_coast = t_coast  v_final

Stage 1 Notes

v_final = 2384 m/s = 5330 miles per hour; tau selected to match NASA data.

Acceleration at lift-off: a(t=0) = 2.3 m/s² = .23 g’s

a(t_final) = 3.1 g’s  

x(t_final) = 78 miles; reasonable; this would include the 38 mile altitude and some distance downrange.

Stage 1 Results and Graphs

including gravity term  g’ = g cos[(pi/2) t /tau)]

tau = 256 seconds

g’ = 5.94 m/s²

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Page  6

Page  4

This latest result gives the Stage 1 displacement as 78 miles (x(t_final) = 126 km); this is the distance traveled by the rocket.  As the NASA data mentions, at the end of the first stage the rocket is downrange about 58 miles (93 km) with an altitude of 38 miles. The graph below shows a hypothetical trajectory based on natural log, a*ln(b*x+1), with a steep ascent and which has a path length of 78 miles, showing that it is not impossible to fit this result within NASA’s data.  Using the velocity data to estimate the diminishing g, and now matching the path length, I think this has worked reasonably well.

Page 1

Momentum

Derivation of the Rocket Equation

Thrust

 

Page 2

Speed of the Rocket During the Burn

 

Page 3

Rocket Acceleration

Position as a Function of Time

 

Page 4

Gravity

 

Page 5

Saturn V

Data

Stage 1 Calculation

 

Page 6

Saturn V

Stage 2, Stage 3 Calculations

 

Page 7

Saturn V Photo Gallery

 

Page 8

Saturn V Photo Gallery

 

Page 9

Equation Review

1 Rocket Science

2 Speed

3 Acceleration

4 Gravity

5 Saturn V—Stage 1

6 —Stage 2 —Stage 3

7 Saturn V Photos

8 Saturn V Photos

9 Equation Review

H        The Saturn V Rocket

 

The Saturn V was a powerful rocket ultimately used to launch the Apollo spacecraft toward the moon.  It was a three stage rocket which had a total mass of 2.77x10⁶ kg  at lift-off.

 

First stage

The first stage produced 33.4x 10⁶ N  (7.5 million pounds ) of thrust with five engines.  The duration of the burn is 150 seconds .

The fuel was 768,000 liters  (203,000 gallons ) of refined kerosene, and the oxidizer was 1,250,000 liters  (331,000 gallons ) of liquid oxygen (LOX).  This amounts to 0.614x 10⁶ kg  of kerosene and 1.43x 10⁶ kg  of LOX, for a total fuel mass of 2.04x 10⁶ kg . 

The mass of the frame of the first stage was 0.136x 10⁶ kg .  This is jettisoned after the fuel is expended.

4.4 second  coast period between first and second stages

 

Second Stage

Five engines produced a thrust of 4.46x10⁶ N .  The burn duration was 361 seconds .  The fuel was 984,000 liters  (260,000 gallons ) of liquid hydrogen; the oxidizer was 314,000 liters  (83,000 gallons ) of LOX.  The total mass of fuel for this stage was [0.0699x10⁶ kg (LH)+ 0.358x10⁶ kg (LOX)] = 0.428x10⁶ kg .

The frame of the second stage had a mass of 0.0432x10⁶ kg .  This too is jettisoned after its fuel is expended.

6.5 second  coast period between second and third stages

 

Third stage

Single engine stage produced 1.00x10⁶ N of thrust.

Burn time for the entire load of fuel was 477 seconds. This rocket engine would burn for 165 seconds  (2.75 minutes ) not expending all of the fuel.  This puts the rocket into orbit around the earth, "Earth Parking Orbit" (EPO) for up to three revolutions while system checkout is performed.  Then, if all systems are "go" the remaining fuel would burn for 312 seconds  (5.2 minutes ) putting the rocket into its trajectory towards the moon ("translunar injection").

Total mass of the fuel for this stage was 0.0119x10⁶ kg .

Empty, the frame of the third stage had a mass of 0.0154x10⁶ kg .  In addition to the frame there was a .00204x10⁶ kg  (45,000 pound ) instrument package, which sat atop the third stage, used to control the three stages in flight: ignition and cutoff of the engines, steering, and any other controls for the Saturn V rocket.  This total mass was 0.0174x10⁶ kg .

 

Above the third stage was the Apollo spacecraft.

Stage 1 equations

Stage 1 equations with g