Gravity is essentially a property of mass, and the force of gravitational attraction between two bodies is given by the formula:

In our scenario, the masses remain the same, and of course  is a constant, so the only thing that changes is the denominator.

On earth, .

The law of universal gravitation is equal to .

We can set these equations equal to one another and isolate  by dividing both sides by , the mass of an object on Earth.

Using the mass of the Earth, the radius of the Earth, and the gravitational constant, , we get a value of approximately if we solve for .

One way an object can be considered "weightless," is if it is accelerating downward at the same rate that it would be if it were free falling. This is why objects in an elevator whose cable had been cut would be "weightless." Therefore the answer is just for the block to be accelerating in the same magnitude and direction as the acceleration due to gravity, which is 

We know , and the force of gravity is:

Also, the equation for uniform circular motion, such as a satellite in orbit is:

Set  and substitute the acceleration due to circular motion into the equation. Solve for velocity.

This indicates that the only variables that affect the orbital speed are orbital radius and the mass of the Earth.

The block cannot provide any frictional force to the left because it is resting on a frictionless table. Whether the block is allowed to move or not, the tension force in the string is all generated by the suspended mass. The force in the string is equal and opposite to the gravitational force acting on the suspended mass.

The system is in equilibrium when no parts are in motion. Alternatively stated, the upward tension in the string is equal and opposite to the downwards force on the mass. If the block is not allowed to move as more mass is added, at some point the tensile force in the string will exceed its mechanical limits, and the string will break.

For the problem we need to understand Newton's second law: . The net upward forces and net downward forces must equal the product of mass and acceleration. Since the box is traveling upwards with an acceleration of 1m/s² we can indicate upward forces as positive and downward forces as negative. Indicating T as tension and Fg as the weight of the box, we can find the value of tension with the equation .

Gravitational acceleration is related to distance via the equation:

 is the gravitational constant,  is the mass of the attracting object, and  is the distance from its center.

In this case, the initial distance (on the surface) is from the center of planet X, and the distance of the satellite is  from the center. We know that gravitational acceleration is proportional to the distance squared, and we know the acceleration at the surface. Using these values, we can solve for the acceleration on the satellite.

By expanding the equation for the acceleration on the satellite, we can see that it is equal to one-sixteenth the acceleration at the surface, based on our original equation. Substitute the value of the surface acceleration to get the final answer.

To solve this question, we will need to use the equation for acceleration:

In this case, the initial velocity will be equal to the final velocity, but opposite in magnitude. The initial velocity is in the upward direction, while the final velocity is downward.

Plug this value into the equation for acceleration.

The velocity value is constant, regardless of the planet. Substitute the acceleration for each planet to determine the change in the time variable.

We can see that the time of flight on the moon is equal to six times the time of flight on Earth.

The force due to gravity on any object can be given by the equation below.

 is the gravitational constant,  is the mass of the earth,  is the mass of the object, and  is the distance between the center of each object.

In our question, the only value to change is the radius of the new planet; both masses and  remain constant. The effect of doubling the radius on the force is given below.

The person's weight on the new planet would be one-fourth their weight on Earth.

To relate force and mass, we use Newton's second law:

We are given his weight (force) on the moon, and we are told the relative gravitational acceleration on the moon. Using these values, we can find the astronaut's mass.

Since mass is the same regardless of gravitational acceleration, this is the same as the astronaut's mass on the earth.