Bull. Volcanol. Soc. Japan
Vol. 58 (2013) No. 1, pp. 281-289

ArticleRenovated Ballistic Equation of Ejected Blocks and Its Application
to the 1982 and 1983 Sakurajima Eruptions
Takeo SUZUKI＊, Yasunori NISHIDA＊＊ and Kiyoaki NIIDA＊＊＊(Received September 29, 2011 ; Accepted December 9, 2012）
We propose a ballistic equation of ejected volcanic blocks that is renovated by considering air resistance, windvelocity and direction dependence of the initial velocity. In the equation, we treat the air resistance as a vector thatcoincides with the opposite direction of flight. By comparing the calculated results based on the equation of motion andthe observed data, such as spatial distribution of landed blocks, initial velocities by successive photos and so on, wetried to reproduce kinematic aspects of the ejected blocks at the 1982 and 1983 eruptions of Sakurajima Volcano,
Kyushu, Japan.
Key words : ballistic equation of volcanic blocks, air resistance as a vector, Sakurajima volcano

1．Introduction
Volcano-energetic study is very important for discussing the eruption process. In addition to the energy releasedue to the crustal deformation, seismic, and plume riseactivities, estimate of the volcanic block discharge energy
is important not only for volcanological standpoints but
also for preventing disasters. For these purposes, weneed to evaluate the velocity, direction and mass distribution of the ejected blocks at the exit of a crater. However, in most cases, it is hard to measure directly such
quantities during an eruption. Therefore, the distributionof volcanic blocks landed on and around the crater has
been investigated after the eruptions to reproduce thekinetic aspects of the ejected blocks.
Ballistic curves of volcanic blocks are controlled byeffects such as the direction of the principal axis ofejection, the velocity and direction of the wind, the air
resistance to the blocks and the direction dependence of
the initial velocity of ejection. Among these effects, theair resistance plays an important role for accurately
estimating the initial velocity. Matuzawa (1934) pointed
out that the air resistance applied to a flying object shouldbe treated as a vector. However, previous works (e. g.,
Nagata, 1938; Minakami, 1942; Iguchi et al., 1983; Iguchiand Kamo, 1984) did not take account of this problem.
In the present paper, we numerically calculate syntheticballistic curves of volcanic blocks for various combinations of the above-mentioned parameters including the air＊

Polytechnic University (ret.).
Present address: Midori-ku, Nishihashimoto, 1-17-3-8,Sagamihara, Kanagawa 252-0131, Japan.
＊＊
Hokkaido University (ret.).
Present address: Chuo-ku, N3-W13-3-2-301, Sapporo
060-0003, Japan.

resistance treated as a vector. Referring to these results,
we tried to reproduce detailed kinematic aspects of theejected blocks at the 1982 and 1983 eruptions of Sakurajima Volcano, Kyushu, Japan.
2．Ballistic equation of ejected blocks
In this chapter, we count up some important factors
which affect the ballistic curves of ejected blocks: airresistance, wind velocity, direction of the initial velocity ofejection, inclination of the principal axis of explosion andsize of the blocks.
2-1 Equation of motion considering the air resistanceMatuzawa (1934: in a revised paper of Matuzawa, 1933)
pointed out that the air resistance working to a flying
object should be treated as a vector, although he did notpresent any detailed form. Then, we formulate an equation of the motion of a flying object following theMatuzawaʼs suggestion. In the present paper, we treat intensity of the air resistance is in proportion to the second
power of the block velocity, V 2, while the direction of theapplied force is opposite to that of the flight, −V. Thus,
we propose the equation of motion as:
dVm=m+f ,(1)
dt
where m, V, and g are respectively the mass of block,the flying velocity of block and the acceleration of gravity,respectively. f means air resistance vector. The inertial
air resistance f is expressed as:＊＊＊The Hokkaido University Museum. Kita-ku, N10-W8,
Sapporo 060-0810, Japan.
Corresponding author: Takeo Suzuki
e-mail: szkitakeo@jcom.home.ne.jp