Momentum
There are two attention laws at plan if a gun is fired: attention of drive and attention of energy. Backfire is explained by the law of attention of momentum, and so it is easier to altercate it alone from energy.
The attributes of the backfire action is bent by the force of the accretion gases in the butt aloft the gun (recoil force), which is according and adverse to the force aloft the ejecta. It is aswell bent by the counter-recoil force activated to the gun (e.g. an operators duke or shoulder, or a mount, in the case of a army gun). The backfire force alone acts during the time that the ejecta are still in the butt of the gun. The counter-recoil force is about activated over a assertive time aeon and adds advanced drive to the gun according to the astern drive supplied by the backfire force, in adjustment to accompany the gun to a halt. There are two appropriate cases of adverse backfire force: Free-recoil, in which the time continuance of the counter-recoil force is actual abundant beyond than the continuance of the backfire force, and zero-recoil, in which the counter-recoil force matches the backfire force in consequence and duration. Except for the case of zero-recoil, the counter-recoil force is abate than the backfire force but lasts for a best time. Back the backfire force and the counter-recoil force are not matched, the gun will move rearward, slowing down until it comes to rest. In the zero-recoil case, the two armament are akin and the gun will not move if fired. In a lot of cases, a gun is actual abutting to a free-recoil condition, back the backfire action about lasts abundant best than the time bare to move the ejecta down the barrel. An archetype of abreast zero-recoil would be a gun deeply clamped to a massive or well-anchored table, or authentic from abaft by a massive wall.
The backfire of a firearm, whether ample or small, is a aftereffect of the law of attention of momentum. Bold that the firearm and projectile are both at blow afore firing, afresh their absolute drive is zero. Bold a abreast free-recoil condition, and apathy the gases ejected from the barrel, afresh anon afterwards firing, attention of drive requires that the absolute drive of the firearm and projectile is the aforementioned as before, namely zero. Stating this mathematically:
p_f + p_p = 0\,
where p_f\, is the drive of the firearm and p_p\, is the drive of the projectile. In added words, anon afterwards firing, the drive of the firearm is according and adverse to the drive of the projectile.
Since drive of a physique is authentic as its accumulation assorted by its velocity, we can carbon the aloft blueprint as:
m_f v_f + m_p v_p = 0\,
where:
m_f\, is the accumulation of the firearm
v_f\, is the acceleration of the firearm anon afterwards firing
m_p\, is the accumulation of the projectile
v_p\, is the acceleration of the projectile anon afterwards firing
A force chip over the time aeon during which it acts will crop the drive supplied by that force. The counter-recoil force accept to accumulation abundant drive to the firearm to accompany it to a halt. This agency that:
\int_0^{t_{cr}}F_{cr}(t)\,dt=-m_f v_f=m_p v_p
where:
F_{cr}(t)\, is the counter-recoil force as a action of time (t)
t_{cr}\, is continuance of the counter-recoil force
A agnate blueprint can be accounting for the backfire force on the firearm:
\int_0^{t_r}F_r(t)\,dt=m_f v_f=-m_p v_p
where:
F_r(t)\, is the backfire force as a action of time (t)
t_r\, is continuance of the backfire force
Assuming the armament are somewhat analogously advance out over their corresponding durations, the action for free-recoil is t_r \ll t_f, while for zero-recoil, Fr(t) + Fcr(t) = 0.
edit Angular momentum
For a gun battlefront beneath free-recoil conditions, the force on the gun will not alone force the gun backwards, but will aswell could cause it to circle about its centermost of mass. The torque (τ) on the gun is accustomed by:
\tau=I \frac{d^2\theta}{dt^2}=h F(t)
where h is the erect ambit of the centermost of accumulation of the gun beneath the butt axis, F(t) is the force on the gun due to the accretion gases, according and adverse to the force on the bullet, I is the moment of apathy of the gun about its centermost of mass, and θ is the bend of circling of the butt arbor "up" from its acclimatization at agitation (aim angle). The angular drive of the gun is begin by amalgam this blueprint to obtain:
I \frac{d\theta}{dt}=h\int_0^t F(t)\,dt = h m_g V_g(t)=h m_b V_b(t)
where the adequation of the momenta of the gun and ammo accept been used. The angular circling of the gun as the ammo exits the butt is afresh begin by amalgam again:
I \theta_f=h\int_0^{t_f} m_b V_b\,dt = h m_b L
where θf is the bend aloft the aim bend at which the ammo leaves the barrel, tf is the time of biking of the ammo in the butt and L is the ambit the ammo campaign from its blow position to the tip of the barrel. The bend at which the ammo leaves the butt aloft the aim bend is afresh accustomed by:
\theta_f=\frac{h m_b L}{I}
The drive of the ejected gases will not accord actual abundant to this result, back abundant of the ejected gases avenue the butt afterwards the ammo has larboard the barrel.
edit Energy
A application of activity leads to a altered equation. From Newton's additional law, the activity of a affective physique due to its motion can be declared mathematically from the translational active activity as:
E = {\frac 1 2} m v^2 = \frac{p^2}{2m}
where:
E\, is the translational active energy
m\, is the accumulation of the body
v\, is its velocity
p\, is its drive (mv)
This blueprint is accepted as the "classic statement" and yields a altitude of activity in joules (or foot-pound force in non-SI units). E_t\, is the bulk of plan that can be done by the abashed firearm, firearm system, or projectile because of its motion, and is aswell alleged the translational active energy. In the accoutrements lexicon, the activity of a abashed firearm is alleged acquainted recoil, chargeless recoil, and backfire energy. This aforementioned activity from a projectile in motion is called: cage energy, ammo energy, actual energy. The activity of the projectile at the point of appulse is accepted as the down ambit activity or appulse activity and about will be hardly abate than the cage activity due to wind attrition acting aloft the projectile.
Again bold free-recoil altitude and bold all advanced drive is due to the projectile, the activity of the projectile will be E_p=p_p^2/2m_p and the activity of the firearm due to backfire will be E_f=p_f^2/2m_f. Since, by Newton's third law, pf + pp = 0, it follows that the ratios of the energies is accustomed by:
\frac{E_p}{E_f}=\frac{m_p}{m_f}
The accumulation of the firearm (mf) is about abundant greater than the projectile accumulation (mp) which agency that a lot of of the active activity produced by the battlefront of the firearm is accustomed to the projectile. For example, a burglarize belief 5 pounds battlefront a 150 atom bullet, the backfire activity will be alone 0.43 percent of the absolute active activity developed. In the case of zero-recoil, the firearm will accretion no energy, and the activity of the projectile will be added by 0.43 percent over that of the free-recoil case.
The backfire activity is about captivated by the apparatus which produces the counter-recoil force, and is blown as heat. For a hand-held firearm, the activity is captivated by the shooter's body, creating a baby bulk of heat. For the argosy cannon from the amount above, it will cycle backwards and the backfire activity will be mostly captivated by the abrasion armament in the caster axles and amid the caster and the address accouter and this activity is afresh adapted to heat.
edit Including the ejected gas
The astern drive activated to the firearm is in fact according and adverse to the drive of not alone the projectile, but the ejected gas created by the agitation of the allegation as well. Likewise, the backfire activity accustomed to the firearm will afflicted by the ejected gas. By attention of mass, the accumulation of the ejected gas will be according to the aboriginal accumulation of the propellant. As a asperous approximation, the ejected gas can be advised to accept an able avenue acceleration of αV0 area V0 is the cage acceleration of the projectile and α is about constant. The absolute drive pe of the propellant and projectile will afresh be:
p_e=m_pV_0 + m_g \alpha V_0\,
where: m_g\, is the accumulation of the propellant charge, according to the accumulation of the ejected gas.
This announcement should be commissioned into the announcement for projectile drive in adjustment to access added a added authentic description of the backfire process. The able acceleration may be acclimated in the activity blueprint as well, but back the amount of α acclimated is about defined for the drive equation, the activity ethics acquired may be beneath accurate. The amount of the connected α is about taken to lie amid 1.25 and 1.75. It is mostly abased aloft the blazon of propellant used, but may depend hardly on added things such as the arrangement of the breadth of the butt to its radius.
