Orbital motion：練習問題

A satellite of mass $$m$$ orbits a planet of mass $$M$$ in a circular orbit of radius $$r$$. Which of the following expressions represents the orbital speed of the satellite?

A spacecraft is launched from the surface of a planet of radius \(R\) and mass \(M\). It reaches a stable circular orbit at an altitude equal to the planet's radius (\(R\)) above the surface. What is the ratio of its orbital speed in this orbit to the escape velocity from the planet's surface?

A spacecraft is in a stable circular orbit around Earth. Which of the following statements correctly describes the acceleration of an astronaut inside the spacecraft experiencing apparent weightlessness?

A satellite of mass \(m\) orbits a planet of mass \(M\) in a circular orbit of radius \(R\). It is then transferred to a larger circular orbit of radius \(2R\) using a Hohmann transfer orbit. Let \(E_1\) be the total mechanical energy in the initial circular orbit, \(E_H\) be the total mechanical energy in the Hohmann transfer orbit, and \(E_2\) be the total mechanical energy in the final circular orbit. Which of the following relationships is correct?

What is the gravitational potential energy of a satellite of mass $$m$$ at a distance $$r$$ from the center of a planet of mass $$M$$?

Two satellites, A and B, orbit the same planet. Satellite A has an orbital period of $$T$$ and an orbital radius of $$R$$. If satellite B has an orbital radius of $$4R$$, what is its orbital period in terms of $$T$$?

Derive the expression for the orbital speed $$v$$ of a satellite in a circular orbit of radius $$r$$ around a planet of mass $$M$$, in terms of $$G$$, $$M$$, and $$r$$.

Explain why the total mechanical energy of a satellite in a stable circular orbit around a planet is always negative, and what this negative value signifies.

A space probe of mass $$m = 500\text{ kg}$$ is orbiting the Earth. The mass of the Earth is $$M_E = 5.97 \times 10^{24}\text{ kg}$$ and the universal gravitational constant is $$G = 6.67 \times 10^{-11}\text{ N m}^2\text{ kg}^{-2}$$. The Earth's radius is $$R_E = 6.37 \times 10^6\text{ m}$$.

(a) Calculate the gravitational potential energy of the space probe when it is at an altitude of $$6.0 \times 10^5\text{ m}$$ above the Earth's surface.

(b) If the probe is moved from this altitude to an altitude of $$1.2 \times 10^6\text{ m}$$, what is the change in its gravitational potential energy?

An artificial satellite of mass $$m$$ is orbiting the Earth in a near-circular orbit. Due to residual atmospheric drag, the satellite gradually loses mechanical energy.

(a) Explain why the orbital radius of the satellite decreases as it loses mechanical energy, even though its orbital speed might appear to increase in the short term. (Hint: Consider the relationship between total energy, kinetic energy, and potential energy in orbit).

(b) If the satellite initially has a total mechanical energy of $$-5.0 \times 10^9\text{ J}$$ and eventually crashes into the Earth's surface (radius $$R_E$$) where its gravitational potential energy is $$-6.0 \times 10^9\text{ J}$$, estimate the total energy dissipated by atmospheric drag during its descent. Assume its kinetic energy just before impact is negligible due to air resistance.