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, we
tried 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 release
due to the crustal deformation, seismic, and plume rise
activities, 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; Iguchi
and 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: air
resistance, 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 the
Matuzawaʼs 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

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Takeo SUZUKI, Yasunori NISHIDA and Kiyoaki NIIDA

f =−κ'V 2 V =−κ'V·V,
V(2)

where κ' is the coefficient of the air resistance, that is,
f=κ'V 2. When we select the Cartesian co-ordinate so thatthe earthʼs surface is the plane z=0 and Z-axis is positivein the vertically upward direction, the following relation
holds as:
V=V x i+V y j+V z k and g=−gk,(3)where i, j, k are the unit vectors along X, Y, Z axes,
respectively. Taking κ'/m=κ, the equation of motion isthen governed by:dV=−k−κV·V.
(4)
dt
The detailed expression of the air resistance f was
proposed by Wilson and Huang (1979) as:1
f=κ'V 2= C D ρASV 2,(5)2
where S, V and ρA denote respectively the crosssectional area, velocity of a volcanic block and the air
1
3 C D ρA
asdensity. As κ'=mκ= C D ρAS, κ, becomes2
4 ρ'dsuming a spherical block with a diameter d and a density
ρ'. The altitude dependence of the air density ρA is given
as a function of z as follows (Matuzawa, 1933):

ρA=ρ0 1−



β A
z ,
T0

(6)

where ρ0 and T0 are the air density and the airtemperature of the standard atmosphere at z=0 when thecrater is assumed to exist at the sea level, respectively. β
denotes the vertical temperature gradient of the air. A is
ρ0T 0
 when P0 is the atmospheric pressure atdefined as
P0 β
the sea level. Therefore, coefficient of the air resistance κ
at altitude z is expressed as:κ=λ 1−



β A
z ,
T0

(7)

when λ is given as:
λ=3 C D ρ0
.
4 ρ'd

(8)

Wilson and Huang (1979) obtained an empirical relation
to express CD in eq. (5) through laboratory experiments offalling volcanic fragments as follows:
C D=(24/R e)F 0828+2 1.07−F ,(9)where Re means the Reynolds number (=Vd ρ/η). η
and ρ denote respectively the coefficients of kinematic
viscosity and density of the air. F represents shapeparameter and F=(b+c)/2a when the object is assumed to
be an ellipsoid of which diameters are a, b and c along
three axes. Although Re is less than 103 when volcanic
block of 1 cm in diameter flies in the air with velocity

below 1 m/s, it is reasonable to assume that the second
term is dominant in comparison to the first term for
relatively large volcanic blocks:
C D=2 1.07−F .

(10)Substituting CD into eq. (8),

 

ρ0
3
λ=  1.07−F
.
2
ρ'd
3

(11)

Coefficient 2 in the above equation is given by assuming that block shapes are spherical. Hereafter, expanding
to realistic cases, we express this coefficient in terms of KD
to be determined by analysis of field data or wind tunnel
experiments:
λ=K D  1.07−F ρ'd .
ρ0(12)

Therefore, coefficient of the air resistance κ in eq. (7)becomes:κ=A
K D ρ0  1.07−Fβ
1− (H+z) .
T0
ρ'd





(13)

Here, H and z denote respectively the altitude of thebottom of the crater and the height of a flying blockmeasured from the bottom of the crater. This equation
indicates that the air resistance is inversely proportional toa diameter of block d and proportional to the coefficientKD. In eq. (13), the values with suffix 0 represent those of
the standard atmosphere at the altitude of H.
2-2 Effect of wind velocity on the motion of volcanic blocks
We consider the effect of wind velocity on the motion of
volcanic blocks by referring to Minakami (1942). Let U
and V be respectively the wind velocity vector and theblock velocity vector. Then, the relative velocity vector
W is given by:
W=V−U.

(14)

As we assume that the air resistance is proportional to
the second power of the relative velocity W, the air
resistance vector f is expressed as:
f=−κ'W 2

 W =−κ'W·W.
W(15)

Putting V=Vx i+Vy j+Vz k and U=Ux i+Uy j, andassuming a horizontal wind field, the relative velocity
vector W is represented as:
W=(V x−U x)i+(V y−U y)j+V zk.

(16)

Therefore, we have an equation of motion under thewind velocity U as:m dt =mg−κ'W·W,
dV

(17)

where W= (V x−U x) +(V y−U y) +V 2z .
22m and gRenovated Ballistic Equation of Ejected Blocks and Its Application to the 1982 and 1983 Sakurajima Eruptions

283

denote respectively the mass of block and the accelerationof gravity. Taking κ'/m=κ, as in Section 2-1, eq. (17) isexpressed as:
dV=−k−κW·W.
dt

(18)

By specifically expressing eq. (18) with Cartesian coordinate, we can write the equation of motion by thefollowing simultaneous differential equations:

 
dy
dy
+κW −U =0
dt
dt
dz
dz
+κW +=0
dt
dt
dx
d 2x
+κW
−U x =0
dt
dt 2
2

2

2

2

y



(19)

where W=  (dx/dt−U x) +(dy/dt−U y) +(dz/dt) .
We could calculate the ballistic curve of the ejected blockby integrating these equations numerically by giving initialcondition V0.
2-3 Direction dependence on the initial velocity ofejected blocks
Iguchi et al. (1983) took successive photos during
nighttime to trace trajectories of volcanic blocks ejected by
5 eruptions of Sakurajima Volcano in 1982-1983. The
photo analyses revealed that vertically ejected blocks had
the maximum initial velocity (Vmax). Then, referring to a
study by Штейнбер
гб (1975) on shock waves arisingfrom explosions, they examined the relation between theejection angle (e) of the blocks and the initial velocity (V)normalized to Vmax and found that the maximum initial
velocity for each ejection angle holds the following
empirical relation as: sin
V=V max sin 15 e,

2

2

2

(20)

where ‘e’ is the angle taken from the horizontal plane(Fig. 8 in Iguchi et al., 1983). This equation is presented
by assuming that the explosion principal axis (Fig. 1) isvertical. In the photo analyses, all the data from the 5eruptions are collectively used. However, when we reexamine the relation V/Vmax versus ejection angle for each
eruption, the index number in the above equation seems totake not only a value of 1.5 but also other values as shownin Fig. 2:
Jun. 8, 1982 eruption: 2.5,
Jul. 23, 1982 eruption: 3.0,
Aug. 7, 1982 eruption: 1.5,
Nov. 1, 1982 eruption: 1.5,
Jan. 10, 1983 eruption: 6.0.
Then, taking into consideration in cases of the inclined
explosion principal axis, we revise the definition of thedirection dependence of the initial velocity of ejected
blocks (V0) using a variable index M instead of a constantvalue of 1.5 in eq. (20) as:

Fig. 1. Diagram of the angles showing the direction of
explosion principal axis and the direction of an ejected
block. The direction of the component on the horizontal plane of the initial velocity V0 of an ejected
block is chosen as x-axis. z-axis is zenith direction andx, y and z axes chose Cartesian coordinate system. The
angles in this figure are as follows:
The azimuth of the explosion principal axis is measured clockwise from the north.
Θ: The angle between the explosion principal axis andz-axis.
Ω: The difference between the azimuth of the explosion principal axis and that of an ejected block.
ϕ: The angle between the explosion principal axis andthe ejected direction of a block.
ψ (=90°−Θ): The angle between the explosion principal axis and the horizontal plane.
θ: The angle between the direction of the initial velocity V0 of a block and x-axis.

V 0(ϕ)=V 0max cos M ϕ,(21)

where V0 max is the maximum initial velocity of the block(Suzuki et al., 2008a). We call M the ‘angle dependent parameter’. This parameter is not for volcanological meaning
but only for empirical one. When the initial elevation
angle of an ejected block is θ, the angle between the
directions of the explosion principal axis and the ejected
block, ϕ, is expressed as:
cosϕ=cosψcosΩcosθ+sinψsinθ.

(22)

A representation of the angles (ϕ, ψ, Ω, θ) is shown in
Fig. 1. One should note that the x-axis is taken to coincidewith the horizontal component of the direction of anejected block. Hereafter, for example, a case that the
explosion principal axis has a inclination of 10° fromvertical axis and has a horizontal component to N45° E isexpressed as N45Θ10.

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Takeo SUZUKI, Yasunori NISHIDA and Kiyoaki NIIDA

2-4

Diameter dependence on the initial velocity ofejected blocks
When we assume that the volcanic blocks of various
sizes are thrown from a crater with the same initial velo-

Fig. 2. Normalized initial velocity versus ejection angle
based on the observed data of 5 eruptions in Sakurajima Volcano (Fig. 7 in Iguchi et al., 1983). The observed data are normalized to the maximum initial
velocity of the each eruption. Using different symbols,
the normalized data are shown in this figure respectively. Curves are calculated relations depending onparameter M.

city, the larger size blocks must fly to longer distancesfrom the crater than the smaller blocks, because the air
resistance varies inversely as the size of blocks as
indicated by eq. (13) (Case 1). Therefore, in this case, thelarger size blocks are expected to land dominantly inperipheral area of the spatially dispersed blocks. On theother hand, large size blocks were mixed frequently withinsmaller size group of landed blocks as observed in the1938 eruption of Asama-yama Volcano (Minakami, 1942)
and the 1977 eruption of Usu Volcano (Katsui et al., 1978)as shown in Fig. 3(a) and Fig. 3(b). These evidences may
be an indication that the initial velocity depends on theblock size (Case 2).
We made numerical model calculations using theballistic equation (eq. (9)) in order to interpret the size
distribution of landed blocks as described above. Landing
distance of the volcanic blocks depends on the initial
velocity, discharge angle and size of block. Fig. 4 shows
relations between the maximum landing distance and theinitial velocity for various sizes of blocks. The landing
distance is proportional to the initial velocity and thediameter of blocks, as a matter of course. Note that the air
resistance KD can be almost ignored for the blocks withlanding distance less than several hundred meters.
Case 1: Assuming the constant initial velocity of 250
m/sec, we calculate the landing distance for various
discharge angles to estimate the diameter dependence on
the maximum landing distance. Calculated results revealthat the maximum landing distance increases monotonously with diameter of the blocks as shown in Fig. 5(a).
Case 2: Assuming that the smaller the size of block, theFig. 3. (a) Open squares show the relation between the observed diameter of blocks and landing distances in case of the1938 eruption of Asama-yama Volcano (Minakami, 1942), while the solid curve shows the calculated one. (b) Same as(a) for the 1977 eruption of Usu Volcano (Katsui et al., 1978). Solid curves in (a) and (b) are calculated respectively for
dmax=13.5 m and dmax=1.7 m.

Renovated Ballistic Equation of Ejected Blocks and Its Application to the 1982 and 1983 Sakurajima Eruptions

larger would be the initial velocity, we propose hypothetically a diameter dependent equation of the initial
velocity V0 as shown in eq. (23):  d  .

V 0max (d)=V 0dmin 1−

d



(23)

max

Let V0dmin and B be respectively the initial velocity of aminute block with diameter dmin and the ‘diameter dependent parameterʼ. dmax represents the maximum size ofblock that can acquire the initial velocity; in other words,
blocks with larger diameter than dmax cannot be thrown out
from the crater. Fig. 5 (b) represents the calculated
maximum landing distance in every block size. Blocks
with middle size such as around 2 m in diameter land at thelongest distance; blocks with larger diameter than about 2
m are mixed within the smaller size group. This tendency
may be a proof of diameter dependence of the initial
velocity, although assumed values of B, dmax and V0dmin
are only examples.
2-5 Combination of direction and diameter dependence of the initial velocity
Combining the direction and the diameter dependenceof the initial velocity in eqs. (21) and (23), we obtain thefollowing relation:d  d   cos ϕ

V 0(d, ϕ)=V 0dmin 1−
Fig. 4. Calculated relation between the initial velocity andthe landing distance. Ten cases are shown for no air
resistance KD = 0 (thick solid curve) and KD = 0. 5
(other curves) depending on the diameter of blocks.
Note that the air resistance KD can be ignored for theblocks landed at distances less than several hundred
meter.

285

max

M

(24)

3．Model calculations and their applications to eruptions of Sakurajima Volcano
We make model calculations of the ballistic curve of anejected block, based on the equation of motion shown in
eq. (19) that includes the effects of the air resistance, thewind velocity, the direction and diameter dependence ofFig. 5. (a) Calculated relation between diameter of volcanic blocks and the maximum landing distance, assuming theconstant initial velocity of 250 m/sec. The maximum landing distance increases monotonously with diameter of theblocks. (b) Calculated relation between diameter of volcanic blocks and the maximum landing distance, assuming thediameter dependent initial velocity. Blocks with the middle size such as around 2 m in diameter land at the longest
distance.

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Takeo SUZUKI, Yasunori NISHIDA and Kiyoaki NIIDA

the initial velocity. In addition, the ballistic equation includes many parameters such as KD , F, M, B, dmax etc.
However, it may be very difficult to give constraint of eachparameter because every eruption is probably a combination of different values of the various parameters. In
addition, the observed field data are not necessarily sufficient to estimate the whole parameters: for examples,
only the distribution of the landed blocks was observed insome cases while only initial conditions of direction andvelocity of ejected blocks were observed in other cases.
We try to reproduce the detailed kinematic aspects ofthe ejected blocks at the 1982 and 1983 Sakurajima
eruptions in order to interpret the observed field data. In
calculations, the following constant values of the standard
atmosphere are assumed as:
air density at the sea level: ρ0=1.226 kg/m3,
air temperature at the sea level: T0=288.15 K,
air pressure at the sea level: P0=101325 Pa,
vertical temperature gradient: β=0.0065 K/m.
We take no account of local and seasonal changes of ρ0,
T0 and P0 because the changes observed in the Sakurajima
region by the Kagoshima Local Meteorological Observatory (KLMO) show an insignificant effect less than about
2 % on the air resistance in comparison with that of thestandard atmosphere. Mean aspect ratio of blocks F is assumed as 0.7 which is generally thought value (e.g., Katsui
et al., 1978). Prior to make model calculations for Sakurajima eruptions, we refer to some values of the parameters
estimated from the analyses of the 1938 eruption of
Asama-yama Volcano, the Central Japan, and the 1977
eruption of Usu Volcano, Hokkaido, Japan (Suzuki et al.,2008b) to reduce the degrees of freedom of the otherparameters. The coefficient concerning the air resistance,
KD, is estimated as about 0.5 in both of the eruptions. Thisvalue is thought to be valid for other volcanic eruptions
because the air resistance is the same everywhere. Therefore, we apply this value to the following three case studies
on the Sakurajima eruptions. The mean density of theblocks, ρ', of Sakurajima Volcano was given by Iguchi etal. (1983) as 2.2×103 kg/m3.
Unfortunately, the effect of the diameter dependence tothe initial velocity is not included in the following model
calculations because the size distribution of landed blocks
was not measured after the Sakurajima eruptions (Iguchi etal., 1983; Iguchi and Kamo, 1984). Therefore, we apply
eq. (21) for the initial velocity.
3-1 The Nov. 1, 1982 eruption of Sakurajima Volcano
Sakurajima Volcano erupted on November 1, 1982 from
‘A-craterʼ. Iguchi et al. (1983) estimated the initial maximum velocity of ejected block as 157 m/s with errors
remain less than 10 %, based on the photo analyses. They
also observed the distribution of landed blocks which havea diameter larger than 1 m as shown in Fig. 6 (solid
circles). The altitude (H) of the bottom of A-crater wasFig. 6. Comparison between the observed distribution of
volcanic blocks in the Nov. 1 1982 eruption of Sakurajima Volcano (solid circles) and calculated ones forvarious values of parameters.

830 m (a.s.l.). The mean wind velocity from the surface toan altitude of 2 km measured by KLMO was −9.1 m/sec
for the N-S component (Un) and 12.7 m/sec for the E-W
component (Ue). The northward and eastward components of wind velocity were taken positive.
Applying the trial and error method, we try to reestimate the initial maximum velocity V0max and the angle
dependent parameter M in order to interpret the distribution of the landed blocks that were thrown from Acrater. We use the value of 1 m in diameter of the blockfor calculations. This value is used commonly for thefollowing analyses of the Jan. 10, 1983 and Oct. 7, 1982
eruptions.
First, we calculate for the following three cases,
applying an angle dependent parameter M of 1.5 based on
the observations of Iguchi et al. (1983) and assuming thedifferent initial maximum velocity to evaluate the range of
uncertainty: (case 1) Calculation is made by assuming thatan initial maximum velocity V0max is 141 m/s and the
explosion principal axis is vertical (N00EΘ00); (case 2)
V0max is 149 m/s and the principal axis is vertical (N00EΘ00); (case 3) V0max is 157 m/s and the principal axis isvertical ((N00EΘ00). Θ denotes supplementary angle of ψas shown in Fig. 1. The calculated limit of landing
distance in the case 2 reproduces well the observed
distribution of the landed blocks, while case 1 and case 3
are excluded for candidates of solutions because thecalculated limits of landing distance are too narrow or too
wide compared to the observed distribution of the landed
blocks, respectively (Fig. 6).
Secondly, we calculated the M dependence with the
distribution of the landed blocks when we use the observed

Renovated Ballistic Equation of Ejected Blocks and Its Application to the 1982 and 1983 Sakurajima Eruptions

Fig. 7. Comparison between the observed distribution of
volcanic blocks in the Jan. 10 1983 eruption of Sakurajima Volcano (solid circles) and calculated ones forvarious values of M and the altitude H of the bottom of
B-crater.

V0max of 157 m/s measured by Iguchi et al. (1983). As
shown in Fig. 6, M=2.0 is suitable for interpreting theobserved distribution of the landed blocks (case 4). However, this value of M is in conflict with the observed one asshown in Fig. 2 and described in Section 2-3.
We conclude, therefore, that Sakurajima Volcano predominantly erupted in the vertical direction with an initial
velocity of 149 m/s. This value is inside the range of observation error. The angle dependent parameter M is 1.5.
3-2 The Jan.10, 1983 eruption of Sakurajima Volcano
The Sakurajima Volcano erupted on January 10, 1983
from ‘B-crater’. Iguchi et al. (1983) reported that thealtitude H of the bottom of B-crater was not clearlyobserved but roughly estimated between 700〜800 m because the crater had been filled with the dense volcanicgas. They also reported on the distribution of landed
blocks which is shown in Fig. 7 (solid circles). In addition, they estimated a maximum initial velocity (V0max) asa reliable value of about 135 m/s through the photo analyses. The observed average wind velocity from the surface to an altitude of 2 km was −10.5 m/sec for the northward component (Un) and 6.8 m/sec for the eastward (Ue)component (after KLMO).
We searched for an appropriate value of M to interpretthe distribution of landed blocks using H = 800 m andV0max=135 m/s. As the crater is narrow and is surroundedby steep and vertical wall, we assumed that the explosion
principal axis is vertical (N00E Θ 00). Among trial anderror calculations, Fig. 7 shows the comparison of the287

typical distribution of landed blocks and the calculated
limits of landing distance for three cases: (case 1) M=1.5;(case 2) M=3.0; (case 3) M=6.0. Case 3 cannot reproduce the distribution of landed blocks because thecalculated limit of landing distance is too narrow to satisfy
the observed distribution. The observed outer limit oflanded blocks is distributed in between the calculated ones
for case 1 and case 2.
Next, we investigated the effect of the altitude H to the
distribution of landed blocks because the bottom of the Bcrater was not clearly observed as mentioned above. Then,
we assume the following two cases: (case 4) M=1.5; H=
750 m; (case 5) M = 3. 0; H = 830 m which is the same
altitude as that of A-crater. The calculated results shownin Fig. 7 reveal that case 2 and case 5 are not distinguishable and we prefer case 4 as a suitable model toreproduce the distribution of landed blocks.
In conclusion, the Jan. 10, 1983 eruption of Sakurajima
probably ejected almost vertically with the maximum initial
velocity of 135 m/s at an altitude of about 750 m. The
present analyses also suggest that the angle dependent
parameter M is considerably smaller than the re-examinedvalue of 6.0 described in Section 2.3 and estimated as 1.5.
This discrepancy may be derived from a small number ofdata.
3-3 The Oct. 7, 1982 Eruption of Sakurajima Volcano
The Sakurajima Volcano erupted at A-crater on October 7, 1982. Unfortunately, the initial velocity of eruptions
was not observed for this activity and only the distributionof landed blocks was observed by Iguchi and Kamo (1984)
as shown in Fig. 8 (solid circles). They proposed a model
to reproduce the distribution of landed blocks assumingthat the explosion principal axis has a tilt of 15° from thevertical axis to the direction of N82° E direction (N82EΘ
15) and the maximum initial velocity was 120 m/s.
In addition, we search for other possible combinationsof parameters to reproduce the distribution of blocks under
the conditions of altitude H=830 m and the observed windvelocity Un=−5.0 m/s and Ue=−4.9 m/s (after KLMO).
For example, assuming a combination of the initial
velocity V0max , direction of the explosion principal axis
and the angle dependent parameter M, we calculate the
distribution of blocks for the following cases such as (case1: V0max=126 m/s, the explosion principal axis is N82EΘ
15, M=1.5), (case 2: 140 m/s, N82EΘ13, 3.0), and (case3: 164 m/s, N82EΘ10, 6.0). As a result, any case withvarious combinations of parameters almost satisfies theobserved distribution of blocks as shown in Fig. 8.
Therefore, it may be difficult to estimate the values of theparameters with only the data of the distribution of thelanded blocks.
3-4 Expected information for estimating constrained
value of parameters
We described in the present paper that the motion of the288

Takeo SUZUKI, Yasunori NISHIDA and Kiyoaki NIIDA

Fig. 8. Comparison between the observed distribution of
volcanic blocks at the Oct. 7, 1982 eruption of Sakurajima Volcano (solid circles) and calculated ones forvarious values of parameters.

ejected blocks is controlled by many physical parameterssuch as the initial velocity and direction of explosion, theangle dependent parameter and so on. However, it is often
difficult to make direct observation of these parameters.
Direction and initial velocity of explosion were relatively
well constrained in the case of the November 1, 1982
eruption as described in Section 3-1 because the initialmaximum velocity V0max and the angle dependent
parameter M were measured. In comparison, only roughestimations of these parameters were made in the case of
the Jan. 10, 1983 eruption because the reliability of theangle dependent parameter M was not necessarily assured.
As various combinations of parameters satisfy the
distribution of the landed blocks, appropriate values of thedirection and initial velocity of explosion were notdetermined in the case of the October 7, 1982 eruption.
To overcome the above-mentioned difficulties for estimating the parameters, we propose additional approaches
to be taken during the next eruption: (1) If we could
measure landing angle of the flying block, it would help us
to constrain the combination of parameters. Fig. 9 shows
examples of calculated relations between the landing
angles and the landing distances from A-crater in theeastward direction. The same values of parameters as thecases 1〜3 in Section 3-3 are applied in this calculation.
The calculated three cases are readily distinguishable. In
an actual field, Minakami (1942) obtained the landing
angles of three bombs at the 1938 eruption of Asamayama Volcano by measuring traces of stuck bombs. Thiscase proved several degrees were distinguishable. (2)Systematic investigation of detailed size distribution of thelanded blocks is needed to prove the diameter dependence

Fig. 9. Calculated relation between landing distances andlanding angles for the parameters used in the cases
1〜3 in Section 3-3.

of the initial velocity.
4．Conclusions
We presented an improved equation of motion considering the air resistance, wind velocity, and direction dependence of the initial velocity. It is noteworthy that we treatedthe air resistance as a vector that coincides with the
opposite direction of flight.
Using this equation, we numerically calculated syntheticballistic curves for various combinations of the abovementioned effects to reproduce the motion of the blocks at
the 1982 and 1983 eruptions of Sakurajima Volcano.
Referring to the observed distribution of landed blocks andphoto data of the explosion, we concluded that the
explosion on Nov. 1, 1982 was predominantly in thevertical direction with an initial velocity of around 149 m/s
and the angle dependent parameter M is 1.5. In case of theJan. 10, 1983 eruption, the observed distribution of landed
blocks and the photo data of the explosion were also
provided for analysis. The principal direction of explosion
was probably vertical with a maximum initial velocity of
135 m/s at an altitude of about 750 m. The angle dependent parameter M was estimated as 1. 5 which isconsiderably smaller than the previously observed value of
6.0. This discrepancy may be derived from small numbersof observed data. Unfortunately, it was difficult to estimate the values of the parameters in case of the Oct. 7,
1982 eruption because only the distribution of landed
blocks was observed.

Renovated Ballistic Equation of Ejected Blocks and Its Application to the 1982 and 1983 Sakurajima Eruptions

Acknowledgements
We would like to express our sincere thanks to Dr. T.
Murase for his valuable discussions. We also thank Dr.
Marie Python, Hokkaido University, for crical readng ofEnglish manuscript. We wish to thank Dr. H. Watanabe
and an anonymous referee for their useful suggestions
which helped improve the quality of the text.
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(Editorial handling Masato Iguchi)

放出岩塊の弾道方程式の改良と 1982・1983 年桜島噴火への適用
鈴木建夫・西田泰典・新井田清信

本論文では，火山爆発による放出岩塊の初速度の方位とサイズの依存性を考慮し，空気抵抗をベクトルとして扱う新たな弾道方程式を提案した．本方程式を桜島南岳爆発の岩塊落下地点分布に適用したところ，
1982 年 11 月 1 日，および 1983 年 1 月 10 日の噴火の最大初速度としてそれぞれ，149 m/s と 135 m/s が得ら
れ，いずれもほぼ鉛直方向に爆発したことが推定された．一方，爆発の主軸が傾く 1982 年 10 月 7 日の噴火
では，岩塊分布を説明可能なパラメータの範囲が広いことが分かった．