A scientific probe travels in a stable circular orbit of radius $r$ around a planet of mass $M$ with an orbital speed of $v_{\text{orb}}$. To send the probe on an interplanetary voyage, engineers plan a brief thruster burn along its instantaneous velocity vector to achieve the minimum speed required to escape the planet's gravitational pull completely. Which expression correctly gives the escape speed $v_{\text{esc}}$ in terms of $v_{\text{orb}}$, along with the correct energetic justification?

A) $v_{\text{esc}} = 2v_{\text{orb}}$, because overcoming the planet's gravitational potential energy requires doubling the probe's kinetic energy, which corresponds to doubling its speed.

B) $v_{\text{esc}} = \frac{v_{\text{orb}}}{\sqrt{2}}$, because moving onto an unpowered outbound path allows gravitational potential energy to convert into kinetic energy without added speed.

C) $v_{\text{esc}} = \sqrt{2}v_{\text{orb}}$, because the probe's total mechanical energy must reach at least zero, requiring kinetic energy to equal the magnitude of potential energy.

D) $v_{\text{esc}} = 4v_{\text{orb}}$, because the inverse-square law of gravity dictates that escape velocity scales with the square of the circular orbital velocity.

A scientific probe orbits a spherical, airless asteroid of uniform density $\rho$ in a circular path just skimming its surface. Mission controllers determine the asteroid's average density using only the probe's measured orbital period $T$. Which statement correctly describes how the asteroid's density $\rho$ relates to the orbital period $T$ and radius $R$, and provides the correct physical justification?

A) $\rho$ is directly proportional to $T^2$ and independent of $R$, because a longer orbital period requires a greater central mass density to keep the probe in orbit.

B) $\rho$ is inversely proportional to $T$ and directly proportional to $R$, because orbital speed increases linearly with the radius for a fixed period.

C) $\rho$ is inversely proportional to $T^2$ and directly proportional to $R$, because the gravitational force decreases with the square of the distance.

D) $\rho$ is inversely proportional to $T^2$ and independent of $R$, because the orbital radius cancels out when equating the gravitational force to the required centripetal force.

A scientific probe travels along an unpowered elliptical orbit around a planet. Flight controllers observe that the probe's speed steadily decreases as it moves from periapsis (closest approach) to apoapsis (farthest point). Which statement correctly explains why the probe slows down and whether its angular momentum about the planet's center is conserved during this segment of the orbit?

A) Gravity does zero work because the gravitational force is perpendicular to the velocity at every point along the orbit, and angular momentum decreases as the speed drops.

B) Gravity does positive work as the probe moves away from the planet, while angular momentum is not conserved because both orbital radius and speed change continuously.

C) Gravity does negative work by pulling against the direction of motion, but angular momentum is not conserved because the magnitude of the gravitational force varies with distance.

D) Gravity does negative work because a component of the gravitational force opposes the motion, while angular momentum about the planet's center is conserved because gravity exerts zero net torque.

An uncrewed spacecraft of mass $m$ is moved from an initial circular orbit of radius $r$ to a higher circular orbit of radius $2r$ around a planet of mass $M$. A mission analyst claims that because the spacecraft travels more slowly in the higher orbit, its total mechanical energy must have decreased. How do the spacecraft's kinetic energy, gravitational potential energy, and total mechanical energy actually change between these two orbits?

A) Kinetic energy decreases, gravitational potential energy decreases, and total mechanical energy decreases because orbital speed is lower at greater distances.

B) Kinetic energy decreases while gravitational potential energy increases by twice as much, resulting in a net increase in total mechanical energy.

C) Kinetic energy increases due to thruster burns, causing both gravitational potential energy and total mechanical energy to increase.

D) Kinetic energy decreases while gravitational potential energy increases by the exact same amount, leaving total mechanical energy unchanged.

A space exploration probe is placed into a stable circular orbit of radius $r$ around an uncharted exoplanet. Flight controllers accurately determine the probe's orbital period $T$. Which expression correctly gives the mass of the exoplanet $M$ in terms of the universal gravitational constant $G$, $r$, and $T$, and what explains why the probe's own mass is not required for this determination?

A) $M = \frac{4\pi^2 r^3}{G T^2}$; the probe's mass is not required because both the gravitational force acting on the probe and its inertial resistance to acceleration are directly proportional to its mass.

B) $M = \frac{G T^2}{4\pi^2 r^3}$; the probe's mass is not required because the gravitational constant balances the ratio between the orbital period and the radius.

C) $M = \frac{4\pi^2 r^2}{G T^2}$; the probe's mass is not required because centripetal acceleration scales inversely with the square of the orbital radius.

D) $M = \frac{2\pi r^3}{G T^2}$; the probe's mass is not required because tangential velocity is independent of the central body's gravitational field.

During a discussion on orbital mechanics, a student claims that astronauts aboard the International Space Station float because the station orbits so high above Earth that Earth's gravitational pull has dropped to zero. In reality, at the station's orbital altitude of approximately 400 km, the acceleration due to gravity is roughly $8.7\text{ m/s}^2$. Which statement correctly evaluates the student's claim and explains why astronauts experience weightlessness in orbit?

A) The student is correct because true weightlessness requires the complete absence of gravitational attraction from any nearby massive body.

B) The student is incorrect because atmospheric drag in low-Earth orbit exerts an upward buoyant force that completely cancels Earth's gravitational pull.

C) The student is incorrect because Earth's gravitational pull remains significant, and apparent weightlessness occurs because the station and astronauts accelerate toward Earth at the same rate in free fall.

D) The student is correct because the high orbital speed generates an outward centrifugal force that physically eliminates the gravitational field inside the cabin.

Two research satellites, $S_1$ of mass $m$ and $S_2$ of mass $3m$, orbit Earth along the exact same circular path of radius $R$. A student claims that because satellite $S_2$ has three times the mass of $S_1$, it must travel at a higher orbital speed and undergo three times the centripetal acceleration to maintain its orbit. Based on Newton's law of universal gravitation and the principles of circular motion, which statement correctly evaluates this claim?

A) The claim is fully correct because maintaining a circular orbit requires a centripetal force proportional to mass, which necessitates a proportional increase in speed.

B) The claim is incorrect because both satellites experience the same centripetal acceleration and travel at the same orbital speed, even though $S_2$ experiences three times the gravitational force.

C) The claim is incorrect because the lighter satellite $S_1$ must travel faster than $S_2$ to prevent Earth's gravitational pull from causing its orbit to decay.

D) The claim is partially correct because while both satellites travel at the same orbital speed, $S_2$ requires three times the centripetal acceleration of $S_1$.

An aerospace engineering team evaluates a proposal to place an unpowered telecommunications satellite into an orbit where it would remain permanently stationary directly above a ground tracking station located at 45° N latitude. The proposal assumes that adjusting the orbital altitude to yield an orbital period of exactly 24 hours will keep the satellite fixed over that location. Based on Newton's law of universal gravitation and the requirements of circular orbital motion, which statement correctly evaluates this proposal?

A) The proposal is feasible because any satellite with an orbital period matching Earth's rotational period naturally remains fixed over its designated surface coordinate.

B) The proposal is feasible only if the satellite's mass is increased so that the gravitational force matches the centripetal force required at that specific latitude.

C) The proposal is impossible because a satellite at that orbital altitude would necessarily travel too slowly to match the rotational speed of Earth's surface.

D) The proposal is impossible because Earth's gravitational force acts toward Earth's center of mass, requiring the center of any circular orbit to coincide with Earth's center.

A scientific satellite in low-Earth orbit experiences a very small resistive force from the outermost fringes of the atmosphere, causing its orbital radius to gradually decrease while maintaining an approximately circular path. How do the gravitational force exerted on the satellite by Earth and the satellite's orbital speed compare to their original values after this radius decrease?

A) The gravitational force increases, and the orbital speed decreases.

B) The gravitational force increases, and the orbital speed increases.

C) The gravitational force decreases, and the orbital speed decreases.

D) The gravitational force decreases, and the orbital speed increases.

Two scientific satellites, Alpha and Beta, move in circular orbits around Earth at orbital radii of $r$ and $4r$ measured from the center of Earth. Satellite Alpha completes its orbit with a speed of $v$ and an orbital period of $T$. Assuming gravity is the only force acting on the satellites, what are the orbital speed and orbital period of satellite Beta in terms of $v$ and $T$?

A) Orbital speed is $2v$ and orbital period is $2T$

B) Orbital speed is $v/2$ and orbital period is $4T$

C) Orbital speed is $v/4$ and orbital period is $8T$

D) Orbital speed is $v/2$ and orbital period is $8T$

Orbits and satellites Transitional Reasoning Download PDF

Physics · Forces & Newton's Laws · Physics · Circular Motion & Gravitation

10 Physics practice questions on Orbits and satellites. Questions that build towards applied reasoning.

Questions

1.
A scientific probe travels in a stable circular orbit of radius $r$ around a planet of mass $M$ with an orbital speed of $v_{\text{orb}}$ . To send the probe on an interplanetary voyage, engineers plan a brief thruster burn along its instantaneous velocity vector to achieve the minimum speed required to escape the planet's gravitational pull completely. Which expression correctly gives the escape speed $v_{\text{esc}}$ in terms of $v_{\text{orb}}$ , along with the correct energetic justification?
- A) $v_{\text{esc}} = 2v_{\text{orb}}$ , because overcoming the planet's gravitational potential energy requires doubling the probe's kinetic energy, which corresponds to doubling its speed.
- B) $v_{\text{esc}} = \frac{v_{\text{orb}}}{\sqrt{2}}$ , because moving onto an unpowered outbound path allows gravitational potential energy to convert into kinetic energy without added speed.
- C) $v_{\text{esc}} = \sqrt{2}v_{\text{orb}}$ , because the probe's total mechanical energy must reach at least zero, requiring kinetic energy to equal the magnitude of potential energy.
- D) $v_{\text{esc}} = 4v_{\text{orb}}$ , because the inverse-square law of gravity dictates that escape velocity scales with the square of the circular orbital velocity.
2.
A scientific probe orbits a spherical, airless asteroid of uniform density $\rho$ in a circular path just skimming its surface. Mission controllers determine the asteroid's average density using only the probe's measured orbital period $T$ . Which statement correctly describes how the asteroid's density $\rho$ relates to the orbital period $T$ and radius $R$ , and provides the correct physical justification?
- A) $\rho$ is directly proportional to $T^2$ and independent of $R$ , because a longer orbital period requires a greater central mass density to keep the probe in orbit.
- B) $\rho$ is inversely proportional to $T$ and directly proportional to $R$ , because orbital speed increases linearly with the radius for a fixed period.
- C) $\rho$ is inversely proportional to $T^2$ and directly proportional to $R$ , because the gravitational force decreases with the square of the distance.
- D) $\rho$ is inversely proportional to $T^2$ and independent of $R$ , because the orbital radius cancels out when equating the gravitational force to the required centripetal force.
3.
A scientific probe travels along an unpowered elliptical orbit around a planet. Flight controllers observe that the probe's speed steadily decreases as it moves from periapsis (closest approach) to apoapsis (farthest point). Which statement correctly explains why the probe slows down and whether its angular momentum about the planet's center is conserved during this segment of the orbit?
- A) Gravity does zero work because the gravitational force is perpendicular to the velocity at every point along the orbit, and angular momentum decreases as the speed drops.
- B) Gravity does positive work as the probe moves away from the planet, while angular momentum is not conserved because both orbital radius and speed change continuously.
- C) Gravity does negative work by pulling against the direction of motion, but angular momentum is not conserved because the magnitude of the gravitational force varies with distance.
- D) Gravity does negative work because a component of the gravitational force opposes the motion, while angular momentum about the planet's center is conserved because gravity exerts zero net torque.
4.
An uncrewed spacecraft of mass $m$ is moved from an initial circular orbit of radius $r$ to a higher circular orbit of radius $2r$ around a planet of mass $M$ . A mission analyst claims that because the spacecraft travels more slowly in the higher orbit, its total mechanical energy must have decreased. How do the spacecraft's kinetic energy, gravitational potential energy, and total mechanical energy actually change between these two orbits?
- A) Kinetic energy decreases, gravitational potential energy decreases, and total mechanical energy decreases because orbital speed is lower at greater distances.
- B) Kinetic energy decreases while gravitational potential energy increases by twice as much, resulting in a net increase in total mechanical energy.
- C) Kinetic energy increases due to thruster burns, causing both gravitational potential energy and total mechanical energy to increase.
- D) Kinetic energy decreases while gravitational potential energy increases by the exact same amount, leaving total mechanical energy unchanged.
5.
A space exploration probe is placed into a stable circular orbit of radius $r$ around an uncharted exoplanet. Flight controllers accurately determine the probe's orbital period $T$ . Which expression correctly gives the mass of the exoplanet $M$ in terms of the universal gravitational constant $G$ , $r$ , and $T$ , and what explains why the probe's own mass is not required for this determination?
- A) $M = \frac{4\pi^2 r^3}{G T^2}$ ; the probe's mass is not required because both the gravitational force acting on the probe and its inertial resistance to acceleration are directly proportional to its mass.
- B) $M = \frac{G T^2}{4\pi^2 r^3}$ ; the probe's mass is not required because the gravitational constant balances the ratio between the orbital period and the radius.
- C) $M = \frac{4\pi^2 r^2}{G T^2}$ ; the probe's mass is not required because centripetal acceleration scales inversely with the square of the orbital radius.
- D) $M = \frac{2\pi r^3}{G T^2}$ ; the probe's mass is not required because tangential velocity is independent of the central body's gravitational field.
6.
During a discussion on orbital mechanics, a student claims that astronauts aboard the International Space Station float because the station orbits so high above Earth that Earth's gravitational pull has dropped to zero. In reality, at the station's orbital altitude of approximately 400 km, the acceleration due to gravity is roughly $8.7\text{ m/s}^2$ . Which statement correctly evaluates the student's claim and explains why astronauts experience weightlessness in orbit?
- A) The student is correct because true weightlessness requires the complete absence of gravitational attraction from any nearby massive body.
- B) The student is incorrect because atmospheric drag in low-Earth orbit exerts an upward buoyant force that completely cancels Earth's gravitational pull.
- C) The student is incorrect because Earth's gravitational pull remains significant, and apparent weightlessness occurs because the station and astronauts accelerate toward Earth at the same rate in free fall.
- D) The student is correct because the high orbital speed generates an outward centrifugal force that physically eliminates the gravitational field inside the cabin.
7.
Two research satellites, $S_1$ of mass $m$ and $S_2$ of mass $3m$ , orbit Earth along the exact same circular path of radius $R$ . A student claims that because satellite $S_2$ has three times the mass of $S_1$ , it must travel at a higher orbital speed and undergo three times the centripetal acceleration to maintain its orbit. Based on Newton's law of universal gravitation and the principles of circular motion, which statement correctly evaluates this claim?
- A) The claim is fully correct because maintaining a circular orbit requires a centripetal force proportional to mass, which necessitates a proportional increase in speed.
- B) The claim is incorrect because both satellites experience the same centripetal acceleration and travel at the same orbital speed, even though $S_2$ experiences three times the gravitational force.
- C) The claim is incorrect because the lighter satellite $S_1$ must travel faster than $S_2$ to prevent Earth's gravitational pull from causing its orbit to decay.
- D) The claim is partially correct because while both satellites travel at the same orbital speed, $S_2$ requires three times the centripetal acceleration of $S_1$ .
8.
An aerospace engineering team evaluates a proposal to place an unpowered telecommunications satellite into an orbit where it would remain permanently stationary directly above a ground tracking station located at 45° N latitude. The proposal assumes that adjusting the orbital altitude to yield an orbital period of exactly 24 hours will keep the satellite fixed over that location. Based on Newton's law of universal gravitation and the requirements of circular orbital motion, which statement correctly evaluates this proposal?
- A) The proposal is feasible because any satellite with an orbital period matching Earth's rotational period naturally remains fixed over its designated surface coordinate.
- B) The proposal is feasible only if the satellite's mass is increased so that the gravitational force matches the centripetal force required at that specific latitude.
- C) The proposal is impossible because a satellite at that orbital altitude would necessarily travel too slowly to match the rotational speed of Earth's surface.
- D) The proposal is impossible because Earth's gravitational force acts toward Earth's center of mass, requiring the center of any circular orbit to coincide with Earth's center.
9.
A scientific satellite in low-Earth orbit experiences a very small resistive force from the outermost fringes of the atmosphere, causing its orbital radius to gradually decrease while maintaining an approximately circular path. How do the gravitational force exerted on the satellite by Earth and the satellite's orbital speed compare to their original values after this radius decrease?
- A) The gravitational force increases, and the orbital speed decreases.
- B) The gravitational force increases, and the orbital speed increases.
- C) The gravitational force decreases, and the orbital speed decreases.
- D) The gravitational force decreases, and the orbital speed increases.
10.
Two scientific satellites, Alpha and Beta, move in circular orbits around Earth at orbital radii of $r$ and $4r$ measured from the center of Earth. Satellite Alpha completes its orbit with a speed of $v$ and an orbital period of $T$ . Assuming gravity is the only force acting on the satellites, what are the orbital speed and orbital period of satellite Beta in terms of $v$ and $T$ ?
- A) Orbital speed is $2v$ and orbital period is $2T$
- B) Orbital speed is $v/2$ and orbital period is $4T$
- C) Orbital speed is $v/4$ and orbital period is $8T$
- D) Orbital speed is $v/2$ and orbital period is $8T$

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