Gravitational Escape Velocity with Saturn V Rocket
by Ron Kurtus (revised 15 February 2016)
Saturn V was an American rocket used in the NASA space programs. The Saturn-powered flight of Apollo 11 spacecraft to the Moon in 1969 gives a typical scenario of a rocket reaching near the gravitational escape velocity of the Earth at a high altitude.
In Apollo 11, the Saturn V rocket burned through three stages, each reaching higher velocities and altitudes. From the NASA data, you can see the altitude and velocity necessary for orbiting the Earth, as well as the altitude for the escape velocity.
Using calculations of orbital or tangential velocity and the required escape velocity, you can see that projecting the rocket in the direction of the parking orbit was the most efficient means of reaching the higher velocity.
Questions you may have include:
- What are some factors in escaping the Earth's gravitation?
- What are the stages of the Saturn V rocket?
- What are calculations of the orbital and escape velocities?
This lesson will answer those questions. Useful tool: Units Conversion
Escaping Earth's gravitation
It takes a great effort to project an object upward so that it will escape the Earth's gravitation.
The object—usually a rocket—must accelerate from zero to over 32,000 kilometers per hour (km/h) or 20,000 miles per hour (mph). This requires traveling to high altitudes before the speed is attained. At lower altitudes, air resistance is a major obstacle. It not only holds back the rocket, but it can also cause overheating at high velocities.
The often-quoted surface escape velocity from gravity of 11.2 km/s is not valid, because it assumes instantaneous acceleration. It also ignores the potential destructive nature from air resistance at such a high velocity and low altitude.
Rockets such as Saturn V typically consist of several sub-rockets or stages, each taking the rocket to a higher velocity and altitude in an effective manner. When the first stage expends its fuel, it separates from the rocket, thus reducing the rocket's weight. Then the second stage ignites, pushing the rocket to even greater heights. Finally, the third stage completes the acceleration.
Saturn V stages
The Saturn V rocket that propelled Apollo 11 to the Moon is an example of reaching near the escape velocity by using three stages.
When Saturn V blasted off from the Earth, the first stage burned for 2.5 minutes, lifting the rocket to an altitude of 68 km (42 miles) and a speed of 2.76 km/s (9,920 km/h or 6,164 mph). This speed was much less than the escape velocity.
After the S-IC stage separated from the Saturn V rocket, the S-II second stage burned for 6 minutes. This propelled the rocket to an altitude of 176 km (109 miles) and a speed of 6.995 km/s (25,182 km/h or 15,647 mph). This speed is close to the orbital velocity for that altitude.
After the S-II stage separated from the rocket, the third stage burned for about 2.5 minutes. It then cut off, and the Apollo 11 went into a "parking orbit" at an altitude of 191.2 km (118.8 miles). Its velocity was 7.791 km/s (28,048 km/h or 17,432 mph).
Parking orbit and required escape velocity
Blasting away from Earth
After several orbits around the Earth, the rocket's engines re-ignited, and it blasted off for what they call translunar injection. According to NASA data, Saturn V reached an altitude of 334.436 km and a speed of 10.423 km/s, at which time the engines were shut down.
This velocity was less than the escape velocity for that altitude (see the Calculations section below), but it was sufficient to take Apollo 11 to the Moon. The gravitational attraction from the Moon facilitated its motion.
Seeking escape velocity
If the attraction from the Moon was not a factor and the purpose of the mission was for Saturn V to reach the escape velocity, the rocket could have blasted off in a vertical direction from the Earth or simply accelerated along the tangential direction or motion.
Going in a vertical direction would require the rocket to accelerate from zero velocity to the escape velocity. However, going up to the escape velocity in the radial direction required much less energy.
Using the data given, you can calculate the orbital and escape velocities.
Parking orbit velocity
At an altitude of 191.2 km, Apollo 11 went into a parking orbit. The stated NASA velocity was 7.791 km/s. Compare this velocity with a calculated orbital velocity:
vT = √(GM/R)
- vT is the tangential orbital velocity in km/s
- G is the Universal Gravitational Constant = 6.674*10−20 km3/kg-s2
- M is the mass of the Earth = 5.974*1024 kg
- R is the radius of the Earth (6371 km) plus the altitude of the rocket
(See Circular Planetary Orbits for more information.)
R = 6371 km + 191.2 km = 6562.2 km
vT = √(6.674*10−20*5.974*1024/6562.2) km/s
vT = √(60.759) km/s
vT = 7.795 km/s
Since the mass and radius of the Earth are approximations, this calculated value for the orbital velocity is sufficiently close to the measured velocity.
Gravitational escape velocity
The escape velocity equation is:
ve = − √(2GM/R)
The negative value ve is the radial escape velocity in km/s. It is negative because the direction is away from the center of mass. Since we are comparing the radial and tangential velocities, it is often more convenient to use se, the positive magnitude or escape speed.
(See Convention for Direction with Center of Mass for more information.)
At an altitude of 334.436 km, Apollo 11 had attained a speed of 10.423 km/s. The calculated gravitational escape velocity or speed at that altitude (R = 6705.4 km) is:
se = √(2*6.674*10−20*5.974*1024/6705.4) km/s
se = √(118.920) km/s
se = 10.905 km/s
Rocket did not have to reach escape speed
Since the rocket was going to the Moon, its velocity (10.423 km/s) did not have to be the escape velocity or speed (10.905 km/s) for that altitude, especially since the gravitational force of the Moon also had an effect on the rocket.
Note: Several sources state that the rocket reached the escape velocity of 11.2 km/s. However, that is incorrect, since 11.2 km/s is the surface escape velocity, which does not apply to the high altitude.
It is possible that calculations were made without referring to the NASA data and were done simply assuming the escape velocity was 11.2 km/s.
Direction of rocket
If the rocket was propelled in a vertical or radial direction, it would have to accelerate from 0 km/s in that direction to 10.905 km/s. However, if the rocket went in the tangential direction, it would accelerate from vT to vs. The resulting change in velocity would be 10.905 - 7.795 = 3.11 km/s, which would require much less energy.
This shows that accelerating in the direction of the parking orbit was more efficient way to reach the escape velocity.
The Saturn V rocket was used to power the Apollo 11 spacecraft to the Moon in 1969. The flight shows how a rocket reaches close to the gravitational escape velocity of the Earth at a high altitude.
The Saturn V rocket went through three stages, until it reached an altitude of 191.2 km to go into a parking orbit. The rocket then accelerated to an altitude of 334.436 km, where it was close to the calculated escape velocity of 10.905 km/s.
Calculations for the velocity of a circular orbit at that altitude closely correspond to the NASA-measured velocity. The calculations also show that going in the direction of the parking orbit was a more efficient way to reach the escape velocity.
Shoot for the stars
Resources and references
The following resources can be used for further study on the subject.
Reports/Apollo 17/Saturn V flight evaluation/4 Trajectory - Unofficial Apollo 17 Flight Journal
Translunar Injection - NASA History statistics
Saturn V - Lunar mission launch sequence from Wikipedia
Saturn V Launch Simulation by Robert A. Braeunig
What is escape velocity? - From PhysLink
Escape Velocity - From Wikipedia
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Gravitational Escape Velocity with Saturn V Rocket