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Open-File Report 01-45

A simple calculator of ballistic trajectories for blocks ejected during volcanic eruptions

By Larry G. Mastin

Thumbnail of and link to report PDF (2.1 MB)Introduction

During the past century, numerous observers have described the violent ejection of large blocks and bombs from volcanoes during volcanic explosions. Minakami (1942) mapped the locations of blocks ejected from Asama Volcano during explosions in 1937. He developed a mathematical expression relating initial velocity and trajectory angle of ejected blocks to the ejection distance, taking into account air drag and assuming a constant drag coefficient. In the late 1950’s, Gorshkov (1959) estimated ejection velocities at Bezymianny volcano during its sector-collapse eruption. Wilson (1972) developed the first mathematical algorithm for ballistic trajectories in the volcanological literature (earlier ones had been available for military applications) that considered variations in drag coefficient with Reynolds number. Fagents and Wilson (1993) advanced the method of Wilson (1972) by considering the effect of reduced drag near the vent. From the 1970’s through the 1990’s other papers, too numerous to mention, have estimated volcanic ejection velocities from ballistic blocks. Since the early 1990’s there has been a decrease in the number of published papers that quantify ejection velocities from ballistic trajectories. This decrease has resulted in part from the appreciation that ejection velocities cannot be uniquely determined by ejection distance due to uncertainties in initial trajectory angle and drag force. On the other hand, the decrease in usage has coincided with an increase in the ease with which ballistic calculations can be made, due to the vast improvement in computer power and in the user-friendliness of computers. During the 1970’s, only volcanologists with mathematical acumen or those who could collaborate with applied mathematicians were able to make such estimates. With 21st century computer power, ballistic computation should be available to anyone as a back-of-the-envelope indicator of explosive power; the only factor preventing such usage is the lack of a user-friendly computer program. In this paper, I describe a program that can be used for quick ballistics calculations. The program, Eject!, was written in Microsoft Visual Basic® and operates on any personal computer running Microsoft® Windows 95 or later.


Instructions on installing Eject! version 1.4

  1. Copy the file "" to the hard drive of your Windows-based computer
  2. Unzip the files in ""
  3. Double-click on the file "setup.exe". This will install the software on your computer.
  4. Once it is installed, go to the "start" button, the to "programs." You will see the program in your program list.

First posted November, 2001

Updated November, 2009

For additional information:
Contact CVO 
Volcano Science Center, Cascades Volcano Observatory 
U.S. Geological Survey 
1300 SE Cardinal Court, Building 10, Suite 100 
Vancouver, WA 98683-9589

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Suggested citation:

Mastin, L. G., 2001, A simple calculator of ballistic trajectories for blocks ejected during volcanic eruptions:  U.S. Geological Survey Open-File Report 01-45, 16p,




Model variation

Using Eject!


One table

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