Conservation of Linear Momentum

If all the external forces are zero (Σ_ext = 0) then  which implies that the total momentum

      (1.1)

Note that this is only true if all the external forces are zero. Halliday calls this a closed, isolated system. Another way of stating (1.1) is

Remembering that  is the total momentum of a system of particles (), the conservation equation is

This is a vector equation, so we must always write it out in x, y, or z components.

Example A rifle of mass m_R fires a bullet of mass mB which emerges at a speed of v_Bf . With what speed does the rifle recoil ?

Solution The bullet-rifle system is a closed, isolated system. When the rifle is held at rest the sum of all external forces is zero. Thus momentum is conserved for the bullet (B)-rifle (R) two body system. The total momentum is , so that conservation of momentum is

Now this is a vector equation, so it must be written in terms of components, namely

but there is only motion in the x direction and nothing is happening in the y direction, so let's re-write the x-equation, leaving off the x's as

or

But v_Ri + v_Bi = 0 because before the gun is fired (initial situation) the bullet and gun do not move. After the gun is fired (final situation) they both move. Thus

where the minus sign indicates that the rifle moves in a direction opposite to the bullet.