Newton's Laws Of Motion

§§ APPLICATIONS OF THE PRINCIPLE OF CONSERVATION OF LINEAR MOMENTUM

1) Recoiling of a gun. When a bullet is fired from a gun, the gun recoils i.e. moves in a direction opposite to the direction of motion of the bullet. The recoil velocity of the gun can be
calculated from the principle of conservation of linear momentum.
    Suppose m₁ = mass of bullet,
                   m₂ = mass of gun, 
                   \(\vec v_1\)  = velocity of  the bullet, 
                   \(\vec v_2\)  = velocity of recoil of the gun.
Before firing, the gun and the bullet both, are at rest. Therefore , total linear momentum before firing = 0. Therefore, total linear momentum before firing = 0. The vector sum of linear 
momenta on firing \( m_1 \vec v_1 + m_2 \vec v_2 = 0 \). According to the principle of conservation of linear momentum, total linear momentum after firing should also be zero. 
    \( \therefore m_1 \vec v_1 + m_2 \vec v_2 = 0 \)
    or  \( m_2 \vec v_2 = - m_1 \vec v_1 \) ............................. (25) 
    or \( \vec v_2 = - \frac{{m_1 \vec v_1 }} {{m_2 }} \)  ............................... (26)
The negative sign shows that direction of \(\vec v_2\) is  opposite to the direction of \(\vec v_1\) i.e. the gun recoils. Further, as m₂> >m₁  therefore  \( \vec v_2 < < \vec v_1 \), i.e. velocity of recoil of the gun is much
smaller than the velocity of the bullet . 
    From eq. (26)   \( v_2 \propto \frac{1} {{m_2 }} \), 
It means that a heavier gun will recoil with a smaller velocity and vice-versa. 
Initial K.E of the system is zero, as both the gun and the bullet are at rest. Final K.E. of the system = \( \left( {\frac{1} {2}m_1 v_1^2 + \frac{1} {2}m_2 v_2^2 } \right) > 0 \). Thus K.E of the system increases ( and is not constant). 
If P.E. is assumed to be constant, mechanical energy(=K.E = P.E) will also increase. 
As M.E. is conserved, therefore, chemical energy of gun powder must have been converted into K.E.
While firing the gun must be held tightly to the shoulder. This would save hurting the shoulder. When the gun is held tightly, the body of the shooter and the gun behave as one body. Total mass becomes large and therefore, recoil velocity of the body and the gun becomes too small.