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9.2 Space: 2. A successful rocket launch

Syllabus reference (October 2002 version)
2. Many factors have to be taken into account to achieve a successful rocket launch, maintain a stable orbit and return to Earth	Students learn to: describe the trajectory of an object undergoing projectile motion within the Earth’s gravitational field in terms of horizontal and vertical components describe Galileo’s analysis of projectile motion explain the concept of escape velocity in terms of the: gravitational constant mass and radius of the planet outline Newton‘s concept of escape velocity identify why the term ‘g forces’ is used to explain the forces acting on an astronaut during launch discuss the effect of the Earth‘s orbital motion and its rotational motion on the launch of a rocket analyse the changing acceleration of a rocket during launch in terms of the: Law of Conservation of Momentum forces experienced by astronauts analyse the forces involved in uniform circular motion for a range of objects, including satellites orbiting the Earth compare qualitatively low Earth and geo-stationary orbits define the term orbital velocity and the quantiative and qualitative relationship between orbital velocity, the gravitational constant, mass of the central body, mass of the satellite and the radius of the orbit using Kepler's Law of Periods account for the orbital decay of satellites in low Earth orbit discuss issues associated with safe re-entry into the Earth’s atmosphere and landing on the Earth’s surface identify that there is an optimum angle for re-entry for a manned spacecraft into the Earth’s atmosphere and the consequences of failing to achieve this angle	Students: solve problems and analyse information to calculate the actual velocity of a projectile from its horizontal and vertical components using: perform a first-hand investigation, gather information and analyse data to calculate initial and final velocity, maximum height reached, range, time of flight of a projectile, for a range of situations by using simulations, data loggers and computer analysis identify data sources, gather, analyse and present information on the contribution of one of the following to the development of space exploration: Tsiolkovsky, Oberth, Goddard, Esnault-Pelterie, O‘Neill or von Braun solve problems and analyse information to calculate centripetal force acting on a satellite undergoing uniform circular motion about the Earth using: solve problems and analyse information using:

Extract from Physics Stage 6 Syllabus (Amended October 2002). © Board of Studies, NSW.
[Edit: 18 June 09]

Prior learning: Preliminary modules 8.4 (subsection 1), 8.5 (subsection 4)

Background: This section is concerned with the context of launching a spacecraft, orbiting it around the Earth and the issues of re-entry. It begins by considering projectile motion, within this context, and then proceeds to rocket launches.

solve problems and analyse information to calculate the actual velocity of a projectile from its horizontal and vertical components using:   

• The actual velocity of a projectile can be analysed into two components, perpendicular to each other. In a convenient Earth frame of reference, they are usually the vertical and horizontal components. The vector sum of those components is the actual velocity. Refer to the figure below.
  
  
  Similarly, the velocity of a projectile at any time can be resolved into components as follows:

describe the trajectory of an object undergoing projectile motion within the Earth’s gravitational field in terms of horizontal and vertical components

• The trajectory of a projectile in the Earth’s gravitational field is parabolic, provided that air resistance is ignored and the acceleration due to gravity is uniform.
  • This complex motion can be analysed by considering its horizontal and vertical components at particular instances during the flight. The horizontal motion of the projectile is a constant velocity (air resistance is assumed negligible). Its vertical motion is changing all the time due to the effect of gravity, which causes the projectile to accelerate at 9.8 m s^-2 downwards.

describe Galileo’s analysis of projectile motion

• Galileo postulated that all objects, regardless of their mass, fall at the same rate. In other words, the acceleration due to gravity is the same for all objects. This is only true if air resistance can be ignored.
  • Experiments based on dropping different objects proved inconclusive so Galileo developed a method of rolling balls down highly polished ramps in order to make his comparisons. This enabled him to slow down the motion enough to make more accurate observations of the motion.

perform a first-hand investigation, gather secondary information and analyse data to calculate initial and final velocity, maximum height reached, range, time of flight of a projectile for a range of situations by using simulations, data loggers and computer analysis

• There are many suitable practical experiences that can be performed as first-hand investigations to achieve these measurements and analyses. When gathering secondary information, a variety of sources may be suitable, including the Internet.
  
  An online investigation can be found at Projectile Motion The University of Tennessee, Department of Physics and Astronomy.
  
  Another investigation Projectile Motion is available at the web site of the Physics Education Technology, University of Colorada at Boulder, Colorado, USA.

explain the concept of escape velocity in terms of the:

• gravitational constant
• mass and radius of the planet

• Escape velocity is the initial velocity required by a projectile to rise vertically and just escape the gravitational field of a planet.
  
  During this rise, the projectile’s kinetic energy transforms into gravitational potential energy, so that: E_k initially = E_p finally

• From the expression provided in Background above, it can be seen that the escape velocity depends upon the universal gravitational constant, G, the mass of the planet and the radius of the planet. It does not depend on any intrinsic property of the projectile.The escape velocity for objects leaving the Earth works out to be approximately 40 000 km h^-1.
  
  It is worth noting that this is a hypothetical concept only. It would not be possible to successfully perform such a launch for two reasons.
  1. Trying to travel through the Earth’s dense lower atmosphere at this speed would produce an enormous amount of heat, sufficient to vaporise the projectile.
  2. Any living thing or delicate equipment would be crushed by the enormous g forces created by the process of suddenly being accelerated to 40 000 km h^-1.

outline Newton’s concept of escape velocity

• Newton created a hypothetical scenario as follows. A person climbed a very tall mountain and launched a projectile horizontally from the peak. The projectile follows a parabolic path (see the above discussion relating to projectile motion) before striking the ground. If another projectile were launched faster than the first, then it would travel further before striking the ground. If yet another projectile were launched fast enough, then it should be able to travel right around the Earth because, as it falls, the surface of the Earth curves away from it. The curve of the projectile’s motion would follow that of the earth’s surface and thus not hit it. This projectile would then be in a circular orbit at a fixed height above the earth’s surface. (For an analysis of the forces operating to establish this circular motion, see below.)
  
  
  
• If a projectile is launched still faster, its orbit will stretch out into an elliptical shape. Even faster launch velocities result in the projectile following a parabolic or hyperbolic path away from the Earth, escaping it entirely.

identify why the term ‘g forces’ is used to explain the forces acting on an astronaut during launch

• The term ‘g force’ is used to express apparent weight as a proportion of true weight. True weight cannot be felt as it is a gravitational force-at-a-distance that acts on each atom of a person’s body. Apparent weight is what a person experiences or feels when an external force acts on them to cause a change in their motion (either magnitude or direction or both). These forces are also called inertial forces because they arise from the body’s inertia or resistance to having its motion changed. They can be calculated.

identify data sources, gather, analyse and present information on the contribution of one of the following to the development of space exploration: Tsiolkovsky, Oberth, Goddard, Esnault-Pelterie, O‘Neill or von Braun

• Some useful data sources on rocket pioneers that are available from the Internet are provided below. Decide on the type of information you will collect. Presenting information in chronological sequence would be appropriate.
  
  The evolution of the rocket NASA
  Robert Goddard and his rockets NASA
  The pioneers of rocketry & space travel The ThinkQuest web site
  
  Note that there are several other important rocket pioneers not mentioned in this syllabus point for which information is readily available. Conversely, not all of the pioneers mentioned have a plentiful supply of information available. Nevertheless, there is a great deal of information available on Tsiolkovsky and Goddard, in particular.
  
  
• Gather information from a range of sources. Analyse the information by identifying trends and relationships as well as contradictions in data and information.
  
  
• Select and use an appropriate media to present your data and information. The use of an annotated timeline would be appropriate.

discuss the effect of the Earth's orbital motion and its rotational motion on the launch of a rocket

• The rotation of the Earth on its axis, and the orbital motion of the Earth around the Sun, can be used to provide a launched rocket with a velocity boost. This allows the operators of the rocket to save fuel in achieving the target velocity.
  
  
• Rockets launching into orbit are launched to the east, in the direction of the Earth’s rotation. They therefore receive a 1700 km h^-1 boost toward their target velocity of approximately 30 000 km h^-1 for a low Earth orbit.
  
  
• Rockets that are heading away from the Earth and further into space are not launched until the direction of the Earth in its orbit around the Sun corresponds with the desired direction. The rocket is then launched up to a low Earth orbit, before firing its rockets again to accelerate ahead of the Earth. The velocity boost received from the Earth’s orbital motion in this case is approximately 107 000 km h^-1.

analyse the changing acceleration of a rocket during launch in terms of the:

• Law of Conservation of Momentum
• forces experienced by astronauts
• Law of Conservation of Momentum

• Rocket propulsion is derived from a force pair (as described in Newton’s third law).

• This is an example of how the Law of Conservation of Momentum can be used to analyse rocket motion. The initial momentum of the rocket and its fuel is zero. This sum must be preserved (the Law of Conservation of Momentum). Note that while the mass of fuel burned in a second is much less than the mass of the rocket, the velocity of the hot exhaust gases is much greater than the velocity of the rocket.
  
  Note also that while the left side of the equation remains quite constant during a burn, the terms on the right side are changing. The mass of the rocket is decreasing significantly as the fuel is burned (typically, 90% of a rocket’s mass is fuel). This means that the velocity of the rocket must increase significantly.
  
  
• Forces experienced by astronauts
  
  
  
  Note that two forces act upon an astronaut during launch: the upward thrust (T) as well as the downward weight (W or mg). Newton’s second law can be used to derive a simple expression for acceleration of a rocket that is launched directly up (using the diagram above):
  
• As described above, if the mass of the rocket decreases during flight and the thrust remains constant, the acceleration of the rocket (and astronauts) increases. Thus the force experienced by the astronaut increases. Refer to the graph above to see how the forces change at different times during the flight into orbit around the earth.

analyse the forces involved in uniform circular motion for a range of objects, including satellites orbiting the Earth

• Objects do not perform uniform circular motion unless they are subject to a centripetal force. This is a force that is always perpendicular to the velocity of the object. That force causes the moving object to continually change direction so that it follows a circular path. The centripetal force is always directed toward the centre of the circular motion.
  
  
• The source of the centripetal force for a range of circular motions is listed here.

Circular motion	Source of centripetal force
Ball on a string whirled in a circle	Tension in the string
Car driving around a corner	Friction between the tyres and the road
Satellite orbiting the Earth	Gravitational attraction between the Earth and the satellite

compare qualitatively low Earth and geo-stationary orbits

• A low Earth orbit is an orbit that lies above the Earth’s atmosphere but below the van Allen radiation belts. This means that its altitude is from approximately 250 km to 1 000 km above the surface of the Earth. A satellite in low Earth orbit will need an orbital velocity of approximately 28 000 km h^-1 to maintain the orbit, which gives it an orbital period of approximately 90 minutes. Examples are the space shuttle (altitude 250–400 km) and the Hubble telescope (altitude 600 km).
  
  
• A satellite in a geostationary orbit has an orbital period equal to the Earth’s, that is, it takes one sidereal day (23 hours 56 minutes) to complete an orbit. To achieve this, the satellite must have an altitude of approximately 35 800 km, which places it near the upper edge of the outer van Allen belt. Its orbital velocity is approximately 11 000 km h^-1. From the ground, a satellite in this type of orbit appears fixed in the sky, which makes it especially useful for communications and weather satellites.

define the term orbital velocity and the quantitative and qualitative relationship between orbital velocity, the gravitational constant, mass of the central body, mass of the satellite and the radius of the orbit using Kepler's Law of Periods

• Orbital velocity is the instantaneous speed (magnitude) in the direction indicated by an arrow (directional) drawn as a tangent to the point of interest on the orbital path.
  
  Quantitative relationship: (where d = average distance between centres of the two masses; v = orbital velocity; M = larger mass; ms = smaller mass; G = gravitational constant; T = period of orbit)
  
  Fg = Fc (gravity is the centripetal force)
  
  
  
  Using Kepler’s Law of Periods,
  :
  

solve problems and analyse information to calculate centripetal force acting on a satellite undergoing uniform circular motion about the Earth using: F=mv²/r

• Recall that the centripetal force for a satellite’s orbital motion is provided by the gravitational force of attraction between the satellite and the Earth. Hence, by using Newton’s Law of Universal Gravitation to calculate this gravitational force, the centripetal force can be determined.
  
  Sample problem: The Hubble telescope has a mass of 1 200 kg and orbits at an altitude of 600 km. Calculate the centripetal force acting on this satellite as it performs circular motion around the Earth.
  
  Solution: The centripetal force can be determined by calculating the gravitational force of attraction between the Hubble telescope (m_ht) and the Earth (M):
  
  

solve problems and analyse information using:

• Problems could take the form of absolute calculations or satellite comparisons. The expression below is a more useful form of Kepler’s third law; the law of periods. It can be applied to any orbiting object, e.g. a satellite orbiting a planet or a planet orbiting the Sun. If comparing two objects orbiting the same central body, then the right hand side of the expression has a fixed value and hence:
  
  
  This was the form of the equation used by Kepler when comparing planets to the Earth.

account for the orbital decay of satellites in low Earth orbit

• A satellite in a stable orbit around the Earth possesses a certain amount of mechanical energy, which is the sum of its kinetic energy (due to its high speed) and its gravitational energy (due to its altitude). The lower the altitude of the orbit, the lower the total mechanical energy is.
  
  
• Satellites in low Earth orbit are subject to friction with the sparse outer fringes of the atmosphere. This friction results in a loss of energy. The loss of energy means that this orbit is no longer viable and the satellite drops down to an altitude that corresponds with its new, lower energy. Ironically, the satellite will be moving faster than before (recall that lower orbits require faster orbital velocities) however the extra kinetic energy is derived from the lost potential energy.
  
  
• This is the process of orbital decay, and it is cyclic, as the satellites new lower orbit resides in slightly denser atmosphere, which leads to further friction and loss of energy. The process is not only continuous but speeds up as time goes on.

discuss issues associated with safe re-entry for a manned spacecraft into the Earth’s atmosphere and landing on the Earth’s surface

• Heat: The considerable kinetic and potential energy possessed by an orbiting spacecraft must be lost during re-entry. As the atmosphere decelerates the spacecraft, the energy is converted into a great deal of heat. This heat must be tolerated and/or minimised. The heat can be tolerated by using heat shields that use ablating surfaces (as used on Apollo capsules) or insulating surfaces (as used on the space shuttle). The heat can be minimised by taking longer to re-enter, thereby lengthening the time over which the energy is converted to heat. The space shuttle uses this technique.
  
  
• g forces: The deceleration of a re-entering spacecraft also produces g forces, typically greater than those experienced during launch. High g forces can be better tolerated by reclining the astronaut, so that blood is not forced away from the brain, and by fully supporting the body. The g forces can be minimised by extending the re-entry, slowing the rate of descent. This strategy is employed by the space shuttle.
  
  
• For a time during re-entry, there is a radio blackout caused by overheated air particles ionising as they collide with the spacecraft. This may be a safety issue if contact is needed between the spacecraft and earth at this phase of its flight.
  
  
• Reaching the surface: Even after surviving the issues listed above, the spacecraft must touch down softly onto the surface of the Earth. Several solutions to this problem have been employed, such as first using parachutes and then splashing into ocean, or using many parachutes before crunching onto the ground, or by landing on an air strip (as performed by the space shuttle).

identify that there is an optimum angle for re-entry into the Earth’s atmosphere and the consequences of failing to achieve this angle

• For any given spacecraft wishing to re-enter safely, an optimum angle of re-entry exists. For Apollo capsules this angle was between 5.2° and 7.2°, although this would differ for other spacecraft.
  
  
• If the angle is too shallow then the spacecraft will rebound, due to compression of the atmosphere beneath it.
  
  
• If the angle is too steep then the spacecraft will decelerate too quickly, creating too much heat and burning up the spacecraft.