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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?

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Introduction

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 image00.png where m is mass, and image01.png 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,image09.png 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 image16.png.  Plugging in this equation, the potential energy present in the crater is represented byimage20.png.

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Middle

.165

.0075

.0430

5.67

4.25

2

.0193

.065

.0075

.0380

4.47

3.36

.065

.0075

.0390

4.70

3.53

.115

.0080

.0440

6.35

5.08

.115

.0085

.0430

6.49

5.53

.165

.0085

.0480

8.01

6.81

.165

.0090

.0470

8.19

7.37

3

.0284

.065

.0100

.0390

6.50

6.50

.065

.0100

.0390

6.50

6.50

.115

.0105

.0450

8.96

9.40

.115

.0110

.0430

8.68

9.55

.165

.0110

.0465

10.0

11.0

.165

.0110

.0450

9.44

10.39

4

.0053

.065

.0070

.0286

0.243

1.70

.065

.0065

.0240

1.61

1.049138

.115

.0075

.0310

3.05

2.288454

.115

.0080

.0280

2.73

2.184873

.165

.0100

.0340

5.06

5.0632

.165

.0090

.0300

3.56

3.206309

*The height of drop is not 10, 15 or 20 cm as described in the procedure because the depth of the sand in the container was taken into consideration.

**The volume was generated from the formula image04.png. The units are cm for purposes of readability.

Columns 6 and 7 are processed data, generated from the raw measurements of crater depth and width.

Graphs:

The graphs which compared volume image03.png crater depth to the height of the drop were generated using data from each individual object, in order to ensure that mass was kept constant. The graphs which compared volume image03.png

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Conclusion

image18.png.1 cm. The experiment was also weaker because in order to measure crater depth the object had to be carefully removed first. This inherently weakened the crater, sometimes causing the depth to be slightly changed from what it had initially been.  The experiment was limited by the fact that only two trials were conducted for each height for a respective mass. Three trials would have been better, as a more accurate average could be defined than if only two trials were conducted. This would also help to eliminate outliers better, because there would be two other data points to average with, versus only one.

 If the experiment was repeated, then each of these weaknesses or limitations could be addressed. The inconsistency associated with using my hand could be eliminated by using a clamping system to drop each ball. This would eliminate much of the induced error, as the clamp could be set at a given height without any unsteadiness or inaccuracy. The problems associated with measuring crater depth could be resolved if a set of steel balls of differing mass were used. A magnet could then be used to extract each ball, rather than lifting each one out with my fingers. This would allow for easier and gentler extraction, again helping to eliminate inconsistency. Also, more trials could be conducted, helping the experiment to achieve a more accurate representation of the data. If these changes were implemented then the experiment would become more accurate.  

...read more.

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