if drop down an stone -that it is like a ball- into the slush and if it was soft enough, stone come into the slush in some distance, if throw it away from a certain height to comes into slush and if we want stone to come into slush 2 times more than before, from what hight should we throw it away?

Well we could assume that the force up from the slush on the ball is constant and that the kinetic energy of the ball goes into work, force times depth

F d = (1/2) m v^2 = m g h
if d is twice then h is twice

To determine the height from which you should throw the stone for it to come into the slush 2 times more than before, we can use the concept of potential energy.

Initially, when you drop the stone into the slush from a certain height, it gains potential energy due to its position. This potential energy is then converted into kinetic energy as the stone falls, until it eventually comes to rest in the slush.

When you throw the stone from a certain height, it gains potential energy again due to its new position. The difference in potential energy between the two heights can be used to determine how much higher you need to throw the stone to achieve the desired result.

Let's assume the initial height from which you drop the stone is "h." To make the stone come into the slush 2 times more than before, you need to throw it from a height that provides twice the initial potential energy.

The potential energy of an object is given by the formula:

PE = m * g * h,

where PE is the potential energy, m is the mass of the object, g is the acceleration due to gravity, and h is the height.

Since the mass and acceleration due to gravity are constant, we can ignore them for the purpose of this calculation. So, the potential energy becomes directly proportional to the height:

PE ∝ h.

To achieve twice the potential energy, you need to double the initial height:

2 * h = 2h.

Therefore, you should throw the stone from a height that is twice the initial height, meaning "2h," in order for it to come into the slush 2 times more than before.