Revisiting Recoil

To Explain Recoil Force Using Newtonian Laws Of Movement Is Not As Simplistic As Merely Being The Equal And Opposite Value Of The Bullet's Momentum At Exit

A. The Relevant Terminology

1. Momentum. Momentum is not a force, it is merely a vector quantity and this quantity is not a value of any work being done. A force is the measure of work being done.

2. Recoil. Recoil is a force, so the bullet’s momentum value at exit from the muzzle can not be the measure of the recoil force, therefore the honourable sir Isaac never said this - not even implied it.

3. Impulse. Impulse is force x the time of application of the force. Since mass x velocity = momentum, mass x the change in velocity caused by a force is the change in momentum the body experiences due to the force. Therefore Impulse = Change in momentum over a time period.

4. Collisions. In a collision an object experiences a force for a specific amount of time that results in a change in its momentum.

B. So What Did Newton Actually Say?

1. The concept of recoil is a derivative of Newton’s Second Law of Motion:

(Fnet = m x a) says that the acceleration of an object is directly proportional to the net force acting upon the object and inversely proportional to the mass of the object”.

2. Recoil is based on the physics of a collision:

In a collision, an object experiences a force for a specific amount of time that results in a change in the momentum value of the object. The mass of the object either speeds up or slows down. The impulse experienced by the object equals the change in momentum experienced by the object. Therefore the Newtonian 2nd equation can be written as Force x time = mass x the change in velocity (the “delta” velocity - or Δ v).

3. So indeed the physics of collisions and recoil are governed by the laws of momentum - but not as a direct opposite value to the bullet’s momentum value as it leaves the muzzle as is often proposed. The equation in (A.3) above - the impulse/momentum change equation is an accurate approach to understanding recoil.

C. Example .416 Rigby, 660 mm barrel, 400 gr bullet at 2 419 ft/sec (738 m/sec), barrel time 0.0015 sec.

1. When the bullet exits the muzzle it has achieved a certain momentum value (19.2 kg.m/sec) which is the product of its mass and velocity as shown above.

2. Should the rifle hang on two very long cords the equal and opposite reaction to the bullet accelerating forward in the bore will be the rifle accelerating backwards. The 19.2 kg.m/sec momentum value of the bullet at muzzle exit , seen in isolation does not determine the amount of the backward swing of the rifle. The backward swing started the very moment the bullet started out of the case mouth and started accelerating towards the muzzle.

3. The rate of acceleration of the bullet during the time of its passage through the bore determined the change in the bullet’s velocity from having been zero at primer explosion to 738 metres per second as it leaves the muzzle. This rate of acceleration of the bullet changed the momentum value of the bullet over time. That can be shown in the force equation:

Force = Mass x change in velocity over time, or also written as Force = change of momentum over time.

4. Because Force equals mass times acceleration or deceleration there is no recoil force experienced until something applies an opposing push force to the rifle against this swing; depending on the solidness (time application) of this force it will either merely decelerate, or altogether stop the swing over that time.

5. The mass of the rifle and the shooters shoulder present the net force against the rearward movement of the rifle. This net force is a measure of the recoil because it changes the rearwards momentum of the rifle from whatever figure it had to zero within a certain time.

6. Had the rifle been in a Ledsled with an incompressible butt, the time of application of the force against the rifle’s rearward movement will be the same as the time taken by the bullet to accelerate down the bore. The rifle and the Ledsled butt-stop will experience the highest possible recoil impulse.

7. Had the rifle with an incompressible butt been held by a shooter crowding into the rifle at a low bench, the time of application of the force against the rifle’s rearward movement will be that time taken by the rifle’s mass and the shooters sturdy but not rock hard steady body to stop the rifle’s movement. Evidently this time will be longer than the time of the bullet down the bore and therefore the recoil impulse will be less.

8. Had the rifle been fitted with a soft, compressible Pachmayr butt pad and held by the shooter when standing up and shooting offhand, the time of application of the force against the rifle’s rearward movement will be the time taken by the rifle’s mass and the somewhat resilient “riding” of the recoil by the shooter, and will be appreciably longer than the bullet’s bore time. The recoil impulse will be the least.

So, when two different cartridges with the same weight bullet at the same velocity is fired from two similarly designed rifles as in (6) above, the only variable between the two bullets will be the rate of acceleration of each bullet down the bore after having left that particular case with its own unique volume, propellant volume, the rate of pressure increase, the maximum pressure, and the time of the pressure volume behind the bullet. All the previous variables in two geometrically different cases will respectively impart a uniqe rate of acceleration to each bullet and a unique change in momentum over time. Whichever bullet will have the higher rate of acceleration and therefore the shorter bore time will have the higher recoil impulse.

Example 1: .416 Ruger, case volume 104.3 gr water, 584 mm barrel, 400 gr bullet at 2 420 ft/sec (738 m/sec), 57 426 psi, momentum 19.2 Ns, barrel time: 0.00126 sec.

Example 2: .416 Rigby, case volume 127.8 gr water, 660 mm barrel, 400 gr bullet at 2 419 ft/sec (738 m/sec), 46 800 psi, momentum 19.2 Ns, barrel time 0.0015 sec.

The .416 Ruger with its 23% smaller case and 10,000 psi higher pressure than the Rigby accelerates the bullet at a higher rate than the larger Rigby case does with its lower pressure, larger gas volume and a longer sustained pressure build-up. Due to the higher in-barrel acceleration of the Ruger it has a higher recoil impulse for the same bullet momentum at exit.

I have experience with regular use of a CZ-550 in .416 Rigby and 400 gr bullets at 2 400 ft/sec in Mozambique over a period of three months. It is a most pleasing cartridge-rifle combination. Recoil impulse is less than a .375 H&H and you can shoot it all day. A lady PH friend shoots exactly the same rifle, using the same bullets I used in Mozambique and she also has zero recoil issue.