Recoil Calculation Example

Quote of the Day

You have not yet begun to consider what sorts of people are these Athenians whom you may have to fight.

— Thucydides, describing a statement by an ambassador from Corinth who was speaking to the Spartan Assembly. The Spartans were bent on war. This quote is similar to Yamamoto's often cited (but unconfirmed) warning to the Japanese government about going to war with the United States.

Introduction

Figure 1: A-10 Firing its GAU-8. (Source)

I have been reading about the US Air Force's battle to retire the A-10 Warthog (Figure 1). The USAF has never cared for the A-10 and has made a number of attempts to replace it with either the F-16 or the F-35. During my reading, I saw the following statement about the recoil of its 30 mm Gatling gun, and the impact of this recoil on the A-10's speed.

The average recoil force of the GAU-8/A is 10,000 pounds-force (45 kN), which is slightly more than the output of one of the A-10's two TF34 engines (9,065 lbf / 40.3 kN each). While this recoil force is significant, in practice a cannon fire burst only slows the aircraft a few miles per hour in level flight.

In this post, I will examine these two statements mathematically to determine if I understand them.

Background

Key Performance Parameters

Figure 2 shows the GAU-8 data as stated on the General Dynamics web page.

Figure 2: GAU-8 Key Performance Parameters. (Source)

GAU-8 and Its Projectile

Figure 3: Three Types of 30 mm GAU-8 Rounds. (Source)

Figure 3 shows the GAU-8's 30 mm projectile. For this exercise, I will assume the projectile has the following characteristics:

• Projectile velocity: v_Muzzle = 3400 feet per second (fps)
• Projectile mass: m_GAU8 = 395 grams
• Rate of fire: r = 6000 round per minute

The GAU-8 can be programmed for different rates of fire. I will assume a 6000 rounds per minute for the maximum rate of fire, which will generate the maximum recoil. I will also assume that the gun is fired in burst of 100 rounds, a number that I am guessing based on the ammunition capacity of 511 rounds. Effectively, I am assuming that the gun only has five bursts available.

To estimate the impact of firing the GAU-8 on the speed of the A-10, I will assume that the A-10 weighs 51,000 pounds, which is its listed maximum takeoff weight.

Analysis

Shortcomings of this Analysis

No explosion-driven device is 100% efficient at converting chemical energy into projectile energy. In the case of gun, it is common to assume that as much as 20% of available powder energy goes into the gases that escape from the end of the barrel. I do not know the impact of this gas discharge on the overall recoil for a GAU-8, but it is significant. I will ignore this escaping gas in my analysis below, which means that my calculations provide a lower bound on the recoil of this weapon.

In reality, recoil can only be accurately estimated with detailed knowledge of the gas discharge characteristics.

Recoil Analysis

Figure 4 shows how to estimate the amount of recoil by assume that recoil is do to the change in momentum caused by the opposing momentum of the fired projectiles. Note that this estimate ignores the momentum of the expelled gases.

Figure 4: Recoil Calculation.

We see that the recoil must be greater than 9,200 pounds, which means that the stated recoil force of 10,000 pounds is reasonable.

Impact on A-10 Speed

Figure 5 shows how you can estimate the reduction in the aircraft's speed caused by the firing of the GAU-8.  I calculate that the impact of the GAU-8 on the A-10's speed is ~4 miles per hours, which roughly agrees with the statement quoted in this post's introduction, i.e. a few miles per hour.

Figure 5: Impact of GAU-8 Firing on A-10 Velocity.

Conclusion

The fact that the GAU-8 has ~5 tons of recoil force is amazing. It is hard to believe a weapon like that can be mounted on an aircraft.

As I worked on this problem, I recalled a Woody Woodpecker cartoon within the movie Destination Moon that illustrated how a firing a rifle can generate thrust.