Final answer:
The size of the crater left by a falling object in sand is related to the object's kinetic energy, which is directly proportional to the square of its velocity. A higher velocity upon impact means a larger displacement of sand and a larger crater, due to the greater kinetic energy transferred.
Step-by-step explanation:
When an object falls into sand and leaves a crater behind, the size of the crater is related to the energy of the falling object. The kinetic energy of an object, which is the energy it possesses due to its motion, is given by the equation KE = 1/2 m v^2, where m is the mass and v is the velocity of the object. Since kinetic energy is proportional to the square of the velocity, an object falling at a higher velocity will have significantly more kinetic energy than one at a lower velocity. For example, if an object's velocity doubles, its kinetic energy increases by a factor of four. This increased kinetic energy upon impact means that the falling object will displace more sand and create a larger crater. Moreover, the potential energy that is converted to kinetic energy as the object falls contributes to the resultant crater's size.
The phenomenon has practical implications, such as understanding the effects of meteorite impacts or the safety considerations in designing structures to absorb impact energy effectively. The principles governing the relationship between impact craters and energy can be used in various scientific and engineering applications.