The relationship between force and acceleration is Newton's second law:

We know the masses, but first we need to find the forces in order to draw a conclusion about the satellites' accelerations. For this calculation, use the law of universal gravitation:

We can write this equation in terms of each object:

We know that the force applied to each object will be equal, so we can set these equations equal to each other.

We know that the second object is twice the mass of the first.

We can substitute for the acceleration to simplify.

The acceleration for  is twice the acceleration for ; thus, the lighter mass will have the greater acceleration.

To find the relationship described in the question, we need to use the law of universal gravitation:

The question suggests that we know the radius and one of the masses, and asks us to solve for the other mass.

Since  is a constant, if we know the mass of the astronaut and the radius of the planet, all we need is the force due to gravity to solve for the mass of the planet. According to Newton's third law, the force of the planet on the astronaut will be equal and opposite to the force of the astronaut on the planet; thus, knowing her force on the planet will allows us to solve the equation.

For this comparison, we can use the law of universal gravitation and Newton's second law:

We know that the force due to gravity on Earth is equal to . We can use this to set the two force equations equal to one another.

Notice that the mass cancels out from both sides.

This equation sets up the value of acceleration due to gravity on Earth.

The new planet has a radius equal to twice that of Earth. That means it has a radius of . It has the same mass as Earth, . Using these variables, we can set up an equation for the acceleration due to gravity on the new planet.

Expand this equation to compare it to the acceleration of gravity on Earth.

We had previously solved for the gravity on Earth:

We can substitute this into the new acceleration equation:

The acceleration due to gravity on this new planet will be one quarter of what it would be on Earth.

For this comparison, we can use the law of universal gravitation and Newton's second law:

We know that the force due to gravity on Earth is equal to . We can use this to set the two force equations equal to one another.

Notice that the mass cancels out from both sides.

This equation sets up the value of acceleration due to gravity on Earth.

The new planet has a mass equal to twelve times that of Earth. That means it has a mass of . It has the same radius as Earth, . Using these variables, we can set up an equation for the acceleration due to gravity on the new planet.

We had previously solved for the gravity on Earth:

We can substitute this into the new acceleration equation:

The acceleration due to gravity on this new planet will be twelve times what it would be on Earth.

For this comparison, we can use the law of universal gravitation and Newton's second law:

We know that the force due to gravity on Earth is equal to . We can use this to set the two force equations equal to one another.

Notice that the mass cancels out from both sides.

This equation sets up the value of acceleration due to gravity on Earth.

The new planet has a mass equal to three times that of Earth. That means it has a mass of . It has the same radius as Earth, . Using these variables, we can set up an equation for the acceleration due to gravity on the new planet.

We had previously solved for the gravity on Earth:

We can substitute this into the new acceleration equation:

The acceleration due to gravity on this new planet will be three times what it would be on Earth.

For this comparison, we can use the law of universal gravitation and Newton's second law:

We know that the force due to gravity on Earth is equal to . We can use this to set the two force equations equal to one another.

Notice that the mass cancels out from both sides.

This equation sets up the value of acceleration due to gravity on Earth.

The new planet has a radius equal to six times that of Earth. That means it has a radius of . It has the same mass as Earth, . Using these variables, we can set up an equation for the acceleration due to gravity on the new planet.

Expand this equation to compare it to the acceleration of gravity on Earth.

We had previously solved for the gravity on Earth:

We can substitute this into the new acceleration equation:

For this comparison, we can use the law of universal gravitation and Newton's second law:

We know that the force due to gravity on Earth is equal to . We can use this to set the two force equations equal to one another.

Notice that the mass cancels out from both sides.

This equation sets up the value of acceleration due to gravity on Earth.

The new planet has a mass equal to six times that of Earth. That means it has a mass of . It also has four times the radius of Earth, . Using these variables, we can set up an equation for the acceleration due to gravity on the new planet.

Expand this equation in order to combine the non-variable terms.

We had previously solved for the gravity on Earth:

We can substitute this into the new acceleration equation:

For this comparison, we can use the law of universal gravitation and Newton's second law:

We know that the force due to gravity on Earth is equal to . We can use this to set the two force equations equal to one another.

Notice that the mass cancels out from both sides.

This equation sets up the value of acceleration due to gravity on Earth.

The new planet has a mass equal to twice that of Earth. That means it has a mass of . It also has half the radius of Earth, . Using these variables, we can set up an equation for the acceleration due to gravity on the new planet.

Expand this equation in order to combine the non-variable terms.

We had previously solved for the gravity on Earth:

We can substitute this into the new acceleration equation:

The acceleration due to gravity on this new planet will be eight times what it would be on Earth.

For this comparison we can use the law of universal gravitation:

The astronaut notices that the force due to gravity on this planet is half what it is on Earth. We can thus relate the force equations for the two planets.

refers to any arbitrary mass on the surface of the planet, and will be constant. The radii of the two planets are equal, and the gravitational constant is also equal. These three variables can cancel out from both sides of the equation.

This relation shows us that the mass of the new planet will be half that of Earth.

For this comparison we can use the law of universal gravitation:

The astronaut notices that the force due to gravity on this planet is twice what it is on Earth. We can thus relate the force equations for the two planets.

refers to any arbitrary mass on the surface of the planet, and will be constant. The radii of the two planets are equal, and the gravitational constant is also equal. These three variables can cancel out from both sides of the equation.

This relation shows us that the mass of the new planet will be twice that of Earth.