Using:

The escape velocity is the velocity necessary to have a kinetic energy that can overcome the gravitational potential. That is, the kinetic energy plus the "big G" gravitational potential energy need to add to zero or a positive number.

Solving for

Plugging in values:

The energy before must be equal to the energy when the object is in motion.

Determine the mass of the object.

Plug into the conservation of energy statement above. 

The final velocity of the object will be .

We must first convert to the proper units.

Now use the equation for kinetic energy.

Plug in and solve for the kinetic energy.

John Elway could throw a football with a kinetic energy of .

For this question we will use conservation of gravitational potential energy.

Plug in and solve for .

The man will gain  of gravitational potential energy by walking to the top of the building.

Use the equation for weight to determine the objects mass.

Plug in and solve for the objects mass.

Use the equation for the potential energy due to gravity.

Plug in and solve for the height.

The bucket can be lifted to a height of .

Using:

Combining the radius and the distance to the surface, converting to and plugging in values:

First, determine the final gravitation potential energy:

Using:

Combining the radius and the distance to the surface, converting to and plugging in values:

Then, determine the initial gravitational potential energy:

Using:

Combining the radius and the distance to the surface, converting to and plugging in values:

There are two forces acting on the coaster at the top of the loop: gravity and centrifugal. It is important to note in this problem the difference between centripetal and centrifugal forces. Centripetal force the is actual force applied to the coaster (points inward) and the centrifugal force is that apparent force felt by the coaster as it travels in a loop (points outward). Just remember that these forces are equal, but in opposite directions. Therefore, we will be using the centrifugal force to calculate the g's felt by the coaster. To do this, we will begin with the expression for conservation of energy:

If we assume that the bottom of the loop has a height of zero, we can eliminate initial potential energy. Also, the only work is done by friction, and it is work done on the surrounds, so it will be negative. Therefore, we get:

Plugging in expressions for each of these, we get:

Rearranging for final velocity, we get:

Where d is half the circumference of the loop:

And the final height is twice the radius:

Plugging these in, we get:

Plugging in our values, we get:

Now we can use this to calculate the centrifugal force:

This is in the opposite direction of the other force acting on the system, gravity. Therefore, we get:

Plugging in expressions we get:

Rearranging for net acceleration, we get:

Then calculating g's we get:

Potential energy is equal to the mass of an object multiplied by gravity multiplied by the height at which the object rests. This is demonstrated by the following equation:

In this problem, you are given the mass and potential energy of the object. Gravity is a constant, therefore there is a single variable left to solve for and this is the height at which the object rests. By plugging in the given values the height is found to be:

The package will need to at least gain  in height, thus it will need to gain

 gravitational potential energy

When it is launched, it will have a kinetic energy equal to

Since energy is conserved:

Solving for