Water Drag Forces Causes An Aircraft’s Body Disintegration when Fall into Water: Case Study of Sriwijaya Air SJ-182 Accident

The crash of Sriwijaya Air flight number SJ-182 in Java sea, January 9, 2021 have paid a lot of attention worldwide. The accident has took 62 lives of passengers and crews. The Boeing 737-500 type aircraft tookoff from the Soekarno-Hatta International airport, Jakarta on 14:36 local time and was lost contact after 4 minutes on air.

The SAR team found the aircraft has disintegrated into debries on the Java sea floor. Why do aircraft disintegrate when they fall into the sea though water is an easily penetrated fluid? The fluid property is very different from solids which are difficulty penetrated.

We will discuss the case of the plane crashing into the sea, specifically  the Sriwijaya Air tragedy because it has attracted a lot of attention, and a number of physical data related to this tragedy are easily found on the internet. We will learn how to mine important data from internet, how to use physics equations to process the data to comes to logical conclusions. We will focus on acceleration to answer the question what is the maximum acceleration allowed for the aircraft to avoid structural damages and how much acceleration of the Sriwijaya Air’s body when entering the water.

It is known in aviation science, a flight-envelope (V-n) diagram to show the load factor as a function of the aircraft speed, as illustrated in Figure 1. The load factor is the ratio of the lift force to the aircraft’s weight. To maintain a safe flight, the flight parameters must be located in the diagram. We conclude that the allowed maximum load factor is about 7 to means the maximum acceleration to avoid structural damage is around seven times the gravitational acceleration. If the acceleration far exceeds \( 7g \) the aircraft could break.

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Which Force Causing Disintegrate?

Now let us estimate the acceleration of the Sriwijaya Air aircraft when it started to touch water surface. We mine the data from the flightaware.com [1] as shown in Table 1. It appears that the plane started experience probles at an altitude of 10,850 feets marked with a descent rate of 4,364 feets/min. During the next 19 seconds, the altitude dropped to 7,075 feets, and the descent rate increased to 11,921 feets / min (a very large descent rate). As comparison, the maximum climbing rate was only 3,900 feets/min. Meanwhile, the descent rate during landing is generally less than 1,000 fets/min. For example, during landing at Soekarno-Hatta International airport, Jakarta, Garuda Indonesia GA507, descent rate was less than 800 feets/min [2].

The descent acceleration of the aircraft depends on the engine thrust, gravity and drag by air. Assume both aircraft’s engines were off and only the drag and gravitation forces acted on the aircraft. The drag force is given by [3]

\( F = {1 \over 2} C A \rho v^2 \)

with \( C \) is the drag coefficient and \( A \) is the cross-section that is perpendicular to the motion, \( \rho \) is the air density, and v is the aircraft speed. The drag coefficient for most passenger aircraft is around 0.03 [4].

To estimate the cross-section, let us take the maximum possibility. We take it as the sum of the wingspan and the maximum body cross-section. For the Boeing 737-500, the wingspan is 91.4 m, the fuselage width 3.76 m, and the total length 31 m [5]. The maximum area (approximating the body as a rectangular) is \( A = 91.4 + 3.76 \times 31 = 208 \) m2.

We need to estimate the speed of the aircraft when touching the water surface. Assume it falls vertically with an initial speed of 11,921 feets/min (60.6 m/s) [1]. The equation of motion under air drag and gravity is \( M dv/dt = Mg – (1/2) C A \rho v^2 \), with \( M \) is the total mass of the aircraft (+passenger+fuel+bagages), \( g \) is the acceleration of gravity.

We momentray define \( w = \sqrt{C A \rho /2Mgv} \) and \( \lambda = \sqrt{C A \rho /2M} \) so that the equation of motion becomes \( dw/dt = \lambda (1-w^2) \). By integration we have [6] \( \ln [(1+w)/(1-w)] = \lambda t + B\), with \( B \) is the constant of integration. The initial condition at \( t = 0 \) is \( v = v_0 = 60.6 \) m/s so that \( B = (1/2) \ln [(1+w_0)/(1-w_0)] \). Using the standard mathematical procedures we obtain

\( v(t) = \sqrt{{2Mg} \over {C A \rho}} \tanh (\lambda t + B) \)

If the time required by the aircraft to reach the water surface is \( T \), the speed is \( v(T) \). This time can be estimated by integrating Eq. (2) to obtain the distance. But the obtained equation is complicated.

To avoid susch a complexity, let us approximate the time \( T \) as the time required for the free fall. Of course, this time is slightly shorter than the true time since the presence of drag. Such a time satisfies \( h = v_0 T + (1/2) g T^2 \). Using \(h \) = 2,156 m (7,075 feets), and \( g = 9.82 \) m/s2 one has \( T \approx 15.7 \) s.

The maximum takeoff wight (MTOW) of the 737-500 is 60,554 kg. Since the aircraft brought 62 passengers [7] out of its maximum capacity of 145 [5], we assume there is a reduction in the mass from its MTOW of about \( (145-62) \times 55 = 4,565 \) kg (assume the average mass of passenger and cabin bagages is 55 kg). Let us take the total mass of about \( 55,000 \) kg. Substituting the corresponding parameter we have \( \lambda = 0.0236 \) s-1, \( B = 0.145 \), and finally \( v(T) \approx 198 \) m/s.

If we calculate using the free-fall equation, the speed when touching the sea surface is \( v(T) = \sqrt{v_0^2 + 2gh} = 215 \) m/s. This value is slightly higher than that obtained by considering air drag, which is a reasonable result.

When the aircraft penetrates the water, it experienced the drag force by water. If the seafloor is shallow, it may also hit the seafloor to produce catastrophic impact. The question is, which force is dominant?

Let us first inspect the acceleration due to water drag. Based on Eq. (1) we obtain \( a = F/m \approx (1/2M) C A \rho v(T)^2 = 2.224 \) m/s2 \( \approx 226 g \). This is a very high acceleration, about \( 226/7 = 32 \) folds of the ultimately allowed acceleration for the aircraft to avoid the structural damage (see Fig. 1). The aircraft will stop after passing a distance \( z = v(T)^2/2a \approx M/CA \rho \approx 8.8 \) meters. The depth of the seafloor at the location where the aircraft debris was found is between 20-23 m [8], 18 meters [9], and 17 m [10]. Based on this data we conclude that, if the aircraft hit the sea in the horizontal position, before reaching the seafloor, the aircraft has stoped and disintegrated into small debris.

Alternatively, if the aircraft hit the sea frontally, the cross-section area is approximately equal to the fuselage cross-section of around 8 m2 (width = 3.76 m and height = 2.139 m) [5]. If this was the condition, the acceleration becomes \( (8/208) \times 2,224 = 85.5 \) m/s2. The aircraft body will stops after passing for around 229 meters to mean that it will hit the seafloor. Let us take the seafloor depth of \( z’ = 20 \) meters and assume the aircfart body stopped on the seafloor, the average acceleration of impact is \( a’ = c^2/2z’ = 980 \) m/s2 \( \approx 98 g \). This accelaration will also disintegrate the aircraft body.

From the above results, we guest two possibilities: if the aircraft penetrated the water in frontal position, the seafloor impact is likely the cause of disintegration. However, if the aircraft impacted the water in a horizontal position, the water drag was responsible for the disintegration.

Based on the last report, the condolences of the pilot and copilot who sat in the from the of aircraft were not identified yet, we believe that the aircraft hit the sea frontally. The front part of the aircraft hit the sea firstly so that the largest damage appeared at the front portion.

References
[1] https://flightaware.com/live/flight/SJY182/history/20210109/0640Z/WIII/WIOO/tracklog
[2] https://flightaware.com/live/flight/GIA507/history/20210115/0815Z/WIOO/WIII/tracklog
[3] Abdullah, M. (2016), Fisika Dasar I (Fundamental Physics Part 1), Bandung Institute of Technology, Bandung.
[4] https://en.wikipedia.org/wiki/Drag_coefficient
[5] https://en.wikipedia.org/wiki/Boeing_737
[6] wolframalpha.com
[7] https://www.inews.id/news/nasional/ini-nama-17-penumpang-sriwijaya-air-yang-teridentifikasi
[8] https://www.suarasurabaya.net/kelanakota/2021/kedalaman-laut-jatuhnya-pesawat-sriwijaya-air-sekitar-20-23-meter/
[9] https://megapolitan.kompas.com/read/2021/01/13/14595901/penyelam-al-kotak-hitam-ditemukan-tertimbun-lumpur-di-kedalaman-18-meter
[10] https://newsmaker.tribunnews.com/2021/01/12/foto-bawah-laut-evakuasi-korban-sriwijaya-air-bikin-merinding-puing-pesawat-uang-baju-bermerek

Feature image source: CNN Indonesia