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Chapter 12: Problem 63

A 1000.-kg communications satellite is released from a space shuttle to initially orbit the Earth at a radius of \(7.00 \cdot 10^{6} \mathrm{~m}\). After being deployed, the satellite's rockets are fired to put it into a higher altitude orbit of radius \(5.00 \cdot 10^{7} \mathrm{~m} .\) What is the minimum mechanical energy supplied by the rockets to effect this change in orbit?

Short Answer

Expert verified
Answer: To find the minimum mechanical energy supplied, first calculate the change in gravitational potential energy (\(\Delta U = U_2 - U_1\)) and assume the change in kinetic energy (\(\Delta K\)) is zero. Then, sum up the changes in gravitational potential energy and kinetic energy to calculate the total mechanical energy change required (\(\Delta E = \Delta U + \Delta K\)). Plug in the values from the given problem to find the minimum mechanical energy supplied by the rockets.

Step by step solution

01

Calculate Initial Gravitational Potential Energy

The gravitational potential energy (GPE) is given by the formula: $$ U = -\frac{GM_Em}{r} $$ where \(G\) is the gravitational constant (\(6.674 \times 10^{-11} N m^2 kg^{-2}\)), \(M_E\) is the mass of the Earth (\(5.972 \times 10^{24} kg\)), \(m\) is the mass of the satellite (\(1000 kg\)), and \(r\) is the distance from the center of the Earth to the satellite. Plug in the values for the initial orbit radius (\(7.00 \cdot 10^{6} m\)) and other constants to calculate the initial GPE: $$ U_1 = -\frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})(1000)}{7.00 \cdot 10^{6}} $$
02

Calculate Final Gravitational Potential Energy

Using the same formula as in step 1, calculate the GPE for the final orbit with a radius of \(5.00 \cdot 10^{7} m\): $$ U_2 = -\frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})(1000)}{5.00 \cdot 10^{7}} $$
03

Calculate Change in Gravitational Potential Energy

Subtract the initial GPE from the final GPE to find the change in gravitational potential energy during the orbit transition: $$ \Delta U = U_2 - U_1 $$
04

Calculate Change in Kinetic Energy

Since the minimum mechanical energy is required, we assume that the change in kinetic energy is zero. This is because the satellite's speed must be kept as low as possible to minimize the required energy input. So, the change in kinetic energy is: $$ \Delta K = 0 $$
05

Calculate the Total Mechanical Energy Change

Finally, sum up the changes in gravitational potential energy and kinetic energy to calculate the total mechanical energy change required: $$ \Delta E = \Delta U + \Delta K $$ Plug the values from steps 3 and 4 into this equation to find the minimum mechanical energy supplied by the rockets to effect the orbit change.

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Most popular questions from this chapter

A satellite of mass \(m\) is in an elliptical orbit (that satisfies Kepler's laws) about a body of mass \(M,\) with \(m\) negligible compared to \(M\) a) Find the total energy of the satellite as a function of its speed, \(v\), and distance, \(r\), from the body it is orbiting. b) At the maximum and minimum distance between the satellite and the body, and only there, the angular momentum is simply related to the speed and distance. Use this relationship and the result of part (a) to obtain a relationship between the extreme distances and the satellite's energy and angular momentum. c) Solve the result of part (b) for the maximum and minimum radii of the orbit in terms of the energy and angular momentum per unit mass of the satellite. d) Transform the results of part (c) into expressions for the semimajor axis, \(a\), and eccentricity of the orbit, \(e\), in terms of the energy and angular momentum per unit mass of the satellite.

a) By what percentage does the gravitational potential energy of the Earth change between perihelion and aphelion? (Assume the Earth's potential energy would be zero if it is moved to a very large distance away from the Sun.) b) By what percentage does the kinetic energy of the Earth change between perihelion and aphelion?

Consider the Sun to be at the origin of an \(x y\) coordinate system. A telescope spots an asteroid in the \(x y\) -plane at a position given by \(\left(2.0 \cdot 10^{11} \mathrm{~m}, 3.0 \cdot 10^{11} \mathrm{~m}\right)\) with a velocity given by \(\left(-9.0 \cdot 10^{3} \mathrm{~m} / \mathrm{s},-7.0 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\right) .\) What will the asteroid's speed and distance from the Sun be at closest approach?

Satellites in low orbit around the Earth lose energy from colliding with the gases of the upper atmosphere, causing them to slowly spiral inward. What happens to their kinetic energy as they fall inward?

Newton was holding an apple of mass \(100 . \mathrm{g}\) and thinking about the gravitational forces exerted on the apple by himself and by the Sun. Calculate the magnitude of the gravitational force acting on the apple due to (a) Newton, (b) the Sun, and (c) the Earth, assuming that the distance from the apple to Newton's center of mass is \(50.0 \mathrm{~cm}\) and Newton's mass is \(80.0 \mathrm{~kg}\).
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