This is a position versus time graph.  To find the velocity of an object at a particular time, determine the slope of the graph at that time.  Using the points  and  we can determine the slope of the graph at 25 seconds since it is a constant slope.

Velocity is equal to a change in displacement per unit time. Consider a basic velocity vs. time graph that depicts a constant velocity, given by a straight horizontal line. How can we determine the displacement during a given time interval? Use the kinematics equation:

Acceleration will be zero, allowing us to reduce the equation. We can also assume that there is zero initial displacement.

So, the displacement will be equal to the velocity multiplied by the time. In the graph, velocity will be represented by the vertical location and time will be represented by the horizontal location of any point. Displacement will be equal to the height of the graph times the width of the graph, or the area of the rectangle created by the constant velocity over a unit of time.

Though this is just one example, displacement will always be equal to the area under the curve of a velocity vs. time graph.

In calculus terms, velocity is the derivative of the function for displacement in terms of time. To find the displacement, one must take the integral of the velocity function. The result is the area under the curve of the velocity function.

This is a position versus time graph.  The slope of the line of the graph tells us the velocity of the object. 

For the first  the object has a constant positive slope indicating that the object is traveling at a constant positive speed.

From  to  the object remains at the same position which means that the object has stopped. 

From  to  the object has a constant negative slope meaning that the object is traveling at a constant negative speed.  

At , the object switches directions and now has a constant positive slope meaning that the object is traveling at a constant positive speed.

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

We know the mass, but we will need to find the force. For this calculation, use the law of universal gravitation:

We are given the value of each mass, the distance (radius), and the gravitational constant. Using these values, we can solve for the force of gravity.

Now that we know the force, we can use this value with the mass of the asteroid to find its acceleration.

The equation for the force of gravity between two objects is:

Using this equation, we can select arbitrary values for our original masses and distance. This will make it easier to solve when these values change.

 is the gravitational constant. Now that we have a term for the initial force of gravity, we can use the changes from the question to find how the force changes.

We can use our first calculation to see the how the force has changed.

We can use the conservation of momentum to solve. Since the satellites stick together, there is only one final velocity term.

We know the masses for both satellites are equal, and the second satellite is initially stationary.

Now we need to find the velocity of the first satellite. Since the satellite is in orbit (circular motion), we need to find the tangential velocity. We can do this by finding the centripetal acceleration from the centripetal force.

Recognize that the force due to gravity of the Earth on the satellite is the same as the centripetal force acting on the satellite. That means .

Solve for  for the satellite. To do this, use the law of universal gravitation.

Remember that r is the distance between the centers of the two objects. That means it will be equal to the radius of the earth PLUS the orbiting distance.

Use the given values for the masses of the objects and distance to solve for the force of gravity.

Now that we know the force, we can find the acceleration. Remember that centripetal force is Fc=m∗ac. Set our two forces equal and solve for the centripetal acceleration.

Now we can find the tangential velocity, using the equation for centripetal acceleration. Again, remember that the radius is equal to the sum of the radius of the Earth and the height of the satellite!

This value is the tangential velocity, or the initial velocity of the first satellite. We can plug this into the equation for conversation of momentum to solve for the final velocity of the two satellites.

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 mg. 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 2r. It has the same mass as Earth, mE. 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.

The formula for force and acceleration is Newton's 2nd law: . We know the mass, but first we need to find the force:

For this equation, use the law of universal gravitation:

We know from the first equation that a force is a mass times an acceleration. That means we can rearrange the equation for universal gravitation to look a bit more like that first equation:

 can turn into:  respectively.

We know that the forces will be equal, so set these two equations equal to each other:

The problem tells us that 

Let's say that   to simplify. 

As you can see, the acceleration for  is twice the acceleration for . Therefore the mass 2m will have the smaller acceleration.

The space station and the astronauts inside are in a constant state of free fall toward the center of the Earth.  However, because they have such a high horizontal velocity and because the Earth is curved they will always be falling toward the earth as the Earth curves away from them.  IF the space station were to slow down, they would land on the Earth.  The high speed in the horizontal direction, keeps them in a parabolic flight path that aligns with the curvature of the Earth.

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 G 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.