How does the mass of a spherical object and the height from which it is dropped into sand affect the width and depth of the crater formed?

Authors Avatar

000345-056

An Investigation of the Formation of Craters

Question: 

How does the mass of a spherical object and the height from which it is dropped into sand affect the width and depth of the crater formed?

Introduction:

        Background Information:

        The formation of a crater is caused by the transfer of kinetic energy from the dropping object to potential energy present in the possible crater. In order to determine the relationship between the mass of the object, the height from which it is dropped from, and the size of the crater the relationship between these two energies must be determined.

The potential energy is equivalent to lifting the crater volume out of the substance and depositing it nearby. This energy can be represented by the equation  where m is mass, and  is the depth of the crater. The mass of the crater is equal to Vd, where d is the density of the crater and V is volume. The crater volume can be defined by the equation for the volume of a spherical cap (a spherical cap is a portion of a sphere which is cut off horizontally). Thus, where a is the radius of the cap and h is the depth. Put into terms of the diameter s, volume can be represented as .  Plugging in this equation, the potential energy present in the crater is represented by.

The kinetic energy comes from the dropping object. At impact this energy is equivalent to the object’s initial potential energy, because it has all been converted into kinetic energy. This is P=mg, where m is mass and  is the height of the drop.

Thus, the relationship between the potential energy present in the crater and the kinetic energy of the dropping object can be represented by  where Y is the energy of the ball and P is the energy of the crater. Then, , or From this, the relationship between the mass of the object, the height from which it is dropped from and the size of the crater can be defined as  because d and g are constants.

Join now!

Hypothesis:

Based on the above relationship, I believe that the more massive the object, or the higher it is dropped from, the larger the resulting crater will be.

Methods:

        Materials:

  • 4 balls of differing mass
  • A Tupperware container (measuring .11 x .20 x .33 m.)
  • Sand
  • A meter stick
  • A scale

        Procedure:

  1. Pour sand into a container to a depth of .035 m.
  2. Put the meter stick on the inside wall of the Tupperware container such that it is oriented vertically with the 1 m mark on the top. ...

This is a preview of the whole essay