Suppose the sand in a sand box with length 14 feet, width 8 feet, and height 1.5 feet is to be removed. The density of the sand (in pounds per cubic feet) h feet from the bottom is given by δ(h)=2.5−h. Find the work required to completely empty the sandbox.

The work required is  

Explanation:

  The volume of the sand box is mathematical represented as

Substituting 14 feet for L,  8 feet for W into the equation

The force as a result of the sand in the disk is mathematically represented as

     Substituting (2.5-h)  for

Now the work that is required to lift the sand from h = 0 to a height of  h=1.5 m is mathematically represented as

Now above is the formula for change in work done in order to obtain the workdone we integrate

here we know that ball will drop down by total distance 3.2 m

so here we know that

now using kinematics

now in the same time it will cover horizontal distance of 0.75 m

so speed of the ball will be

jupiter is the largest planet in our solar system, having a mass and radius that are respectively, 318 and 11.2 times that of earth. suppose that an object falls from rest near the surface of each planet and that the acceleration due to gravity remains constant during the fall. each object falls the same distance before striking the ground. determine the ratio of the time of fall on jupiter to that on earth.