Question:
Explain how Newton’s Laws of Motion and Newton’s Law of Universal Gravitation were applied during the Cassini mission.
Explain how Newton’s Laws of Motion and Newton’s Law of Universal Gravitation were applied during the Cassini mission.
Marking Guidelines:
Criteria
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Marks
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• Explanation using Newton’s Three Laws of Motion and Newton’s Law of Universal Gravitation during the Cassini mission: (1) launch from Earth; (2) travelling to Saturn; and (3) orbiting Saturn.
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6
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Launching the probe from Earth’s surface requires the use of Newton’s third law of motion during the rocket’s operation. This occurs when exhaust gases are expelled downward and thus, there is an upward force on the rocket. During the launch, the rocket’s acceleration is dependent on its mass and the force of the engines as predicted by Newton’s second law of motion.
During the journey to Saturn, the probe does not experience air resistance and continues in its state of motion as described by Newton’s first law of motion.
The Slingshot effect utilizes Newton’s Law of Universal Gravitation as well as the Law of conservation of momentum (Newton’s Third Law of motion) to increase the velocity of the space probe.
A stable orbit can be predicted by using Newton’s Law of Universal Gravitation. The orbital velocity of the space probe is related to its orbital radius.
Based on the marking guidelines, students are expected to explain how Newton’s Law of Universal Gravitation and Newton’s three laws of motion were applied to the three parts of the Cassini mission: launch, traveling to Saturn, orbiting Saturn. However, we can also explain Cassini mission by using the law of conservation of angular momentum and the law of conservation of energy. For instance, Warren (2003), a physics textbook author, writes that “[t]he probe picks up angular momentum from the planet (which loses an equal amount of angular momentum). Gravity allows the ‘coupling’ between the probe and planet to facilitate the transfer. For this reason, gravity-assist trajectories should more correctly be called angular momentum-assist trajectories (p. 30).” In short, Cassini increases speed by using the gravity-assist technique and the planet decreases speed based on the law of conservation of angular momentum or law of conservation of energy.
The sample answer (as mentioned above) with regard to the three parts of the mission: launch from Earth, traveling to Saturn, orbiting Saturn can be improved as follows:
1. Launch from Earth: During the launch of Cassini, the sample answer mentions that the acceleration of the rocket is described by Newton’s second law, and the upward force on the rocket is caused by the exhaust gases that are expelled downward based on Newton’s third law. However, the launch of the probe requires the upward force on the rocket to exceed the weight of the rocket which is governed by Newton’s law of universal gravitation. Furthermore, Newton’s third law can be expressed as a law of conservation of linear momentum such that the increase in forward momentum of the rocket equals to backward momentum of the exhaust gases.
2. Traveling to Saturn: The sample answer states that Newton’s first law of motion is relevant during the Cassini’s journey to Saturn because it does not experience friction or air resistance. On the contrary, Newton’s first law does not strictly apply to the Cassini mission because there are non-zero gravitational forces acting on Cassini everywhere. Importantly, we can explain that the resistive forces due to air resistance and gravitational forces are close to zero when it is reasonably far from the planets.
3. Orbiting Saturn: The sample answer specifies that a stable orbit can be predicted by using Newton’s law of gravitation and the orbital velocity determines the radius of the orbit. However, the orbit of the probe could be elliptical instead of circular based on Kepler’s laws of planetary motion.
Lastly, the sample answer states that the slingshot effect utilizes Newton’s law of gravitation and Newton’s Third Law to increase the velocity of Cassini. Nevertheless, the phrase slingshot effect is a misnomer and it could be replaced by a better term such as gravitational assist or simply gravity assist. However, the technique of gravity assist that increases the speed of Cassini is dependent on the gravitational force of a planet. For example, when Cassini is approaching Jupiter, there is an exchange of orbital kinetic energy and angular momentum between Cassini and Jupiter. Importantly, the total orbital kinetic energy remains constant: Cassini gains orbital kinetic energy whereas the planet loses its orbital kinetic energy.
Feynman’s insights or goofs?:
Firstly, Feynman mentions that “a rocket of large mass, M, ejects a small piece, of mass m, with a terrific velocity V relative to the rocket. After this the rocket, if it were originally standing still, will be moving with a small velocity, v. Using the principle of conservation of momentum, we can calculate this velocity to be v = (m/M)⋅V. So long as material is being ejected, the rocket continues to pick up speed. (Feynman et al., 1963, section 10–4 Momentum and energy).” Note that the velocity of the ejected material (V) is relative to the rocket instead of the Earth. However, to be more precise, we can explain that the initial gain in velocity is (m/M)⋅V, but the subsequent gain in velocity can be increased because of the decrease in the total mass of the rocket, M.
Furthermore, in An Introduction to Mechanics, Kleppner and Kolenkow (2014) write that “[t]he law is often stated in words such as ‘A uniformly moving body continues to move uniformly unless acted on by a force,’ but the underlying concept is really the idea of an isolated body… Newton’s first law raises a number of questions such as what we really mean by an ‘isolated body’ (p. 51).” In other words, there is a problem of defining an isolated body. Similarly, Feynman (1995) explains that “as soon as we allow the presence of gravitating masses anywhere in the universe, concept of such truly unaccelerated motion becomes impossible, because there will be gravitational fields everywhere (p. 93).”
On the other hand, Feynman mentions that “if we can remember some of Kepler’s laws, and add some other laws like the conservation of energy - we can figure out that if the particle didn’t escape, it would make an ellipse, and we can figure out how far away it would get, and that’s what we’re going to do now. If the perihelion of the ellipse is a, how far is the aphelion, b? (Feynman, Gottlieb, & Leighton, 2006, p. 72).” Simply phrased, the probe’s orbital motion is elliptical. More importantly, the terms aphelion and perihelion should be used instead of radius. The aphelion is a point in the orbit of a planet (or a probe) that is furthest from the sun, whereas perihelion is a point in the orbit that is nearest to the sun.
Note:
In Genius: Richard Feynman and modern physics, Gleick (1992) writes that
“Feynman’s spacecraft would use the outer edges of the earth’s atmosphere as a sort of warm-up track and accelerate as it circled the earth. An atomic reactor would power the jet by heating the air that was sucked into the engine. Wings would be used first to provide lift and then, when the speed rose beyond five miles per second, ‘flying upside down to keep you from going off the earth, or rather out of the atmosphere.’ When the craft reached a useful escape velocity, it would fly off at a tangent toward its destination like a rock from a slingshot.
Yes, air resistance, heating the ship, would be a problem. But Feynman thought this could be overcome by delicately adjusting the altitude as the craft sped up—‘if there is enough air to cause appreciable heating by friction there is surely enough to feed the jet engines’ (p. 219).”
“Feynman’s spacecraft would use the outer edges of the earth’s atmosphere as a sort of warm-up track and accelerate as it circled the earth. An atomic reactor would power the jet by heating the air that was sucked into the engine. Wings would be used first to provide lift and then, when the speed rose beyond five miles per second, ‘flying upside down to keep you from going off the earth, or rather out of the atmosphere.’ When the craft reached a useful escape velocity, it would fly off at a tangent toward its destination like a rock from a slingshot.
Yes, air resistance, heating the ship, would be a problem. But Feynman thought this could be overcome by delicately adjusting the altitude as the craft sped up—‘if there is enough air to cause appreciable heating by friction there is surely enough to feed the jet engines’ (p. 219).”
References:
1. Feynman, R. P., Gottlieb, M. A., & Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.
2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.
3. Feynman, R. P., Morinigo, F. B., & Wagner, W. G. (1995). Lectures on gravitation (B. Hatfield, ed.). Reading, MA: Addison-Wesley.
4. Gleick, J. (1992). Genius: Richard Feynman and modern physics. London: Little, Brown, and Company.
5. Kleppner, D., & Kolenkow, R. (2014). An Introduction to Mechanics (2nd ed.). Cambridge: Cambridge University Press.
6. Warren, N. (2003). Excel HSC Physics. Glebe, NSW: Pascal.
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