Momentum

 A quantity that must be taken into account in situations that involve moving bodies is momentum.  Momentum may be though of as a quantitative expression of inertia.  There are two kinds of momentum: linear momentum, which is a measure of the tendency of a moving body to continue in motion along a straight line, and angular momentum, which is a measure of the tendency of a rotating body to continue to spin about an axis that does not change its orientation.

 The linear momentum of a body of mass m and velocity v is defined as the product of m and v.

Note that the product of a scalar and a vector quantity, here m and v respectively, is a vector quantity with the same direction as v and with a magnitude equal to mv.

 Momentum, being a vector quantity, means that is has a direction associated with it, namely the direction of motion of the body.  Kinetic energy, given by the somewhat similar formula: 1/2mv², has a completely different significance, since it is a quantity having only magnitude.  Momentum is conserved in collisions.  When two object collide, the sum of their initial momenta is equal to the momentum of the composite body they form when they stick together.  Momentum is a vector quantity, and the directions of motion of the objects must be taken into account when their momenta are added together.

 Momentum considerations are most useful in situations that involve (in general terms) explosions and collisions.  When external forces do not act upon the bodies involved, their momentum is conserved:  THE TOTAL MOMENTUM OF ALL THE BODIES BEFORE THEY INTERACT IS EXACTLY THE SAME AS THEIR TOTAL MOMENTUM AFTERWARD.  The total kinetic energy of the bodies need not be the same before and after, however; the kinetic energy involved in an explosion is initially zero, since it comes from the chemical energy stored in the explosive material, and the kinetic energy of bodies that collide may disappear, in whole or in part, into heat and sound energy.  In both cases the total momentum does not change.  When two objects collide and stick together, the vector sum of the initial momenta must be equal in magnitude and direction to the momentum of the composite body they form when they stick together.

 Examine what happens when a pistol is fired.  Before the trigger is pulled, both pistol and bullet are stationary, so the initial momentum is zero.  When the trigger is pulled, however, the bullet flies out of the barrel with the momentum m₁v₁, where m₁ is its mass and v₁ its velocity.  Because no outside forces are acting o the bullet and pistol at the moment of firing, the total momentum after firing must be the same as it was initially.  This means that the pistol itself must move backward if its momentum is to balance out the forward momentum of the bullet.  If the mass of the pistol is m₂ and its backward velocity is v₂ then  m₂v₂ = -m₁v₁.    The minus sign indicated that v₂ is opposite in direction to v₁.   It is this momentum that is felt as the recoil of a pistol, rifle, or shotgun; the heavier the bullet and the greater its muzzle velocity, the more ‘kick’ will be felt by the shooter.  In review:  momentum is conserved in explosions.  When a pistol is fired, the backward momentum of the pistol is equal to forward momentum of the bullet.  The speed of the bullet is greater than that of the pistol because it is lighter.  The initial momentum of the system of pistol plus bullet is zero, and the final momentum of the system is also zero.

 The operation of a rocket illustrates conservation of linear momentum.  When the rocket stands stationary on its launching platform, its momentum is zero.  When it is fired, the momentum of the gases rushing out downward is balanced by the momentum in the other direction of the rocket body moving upward; the total momentum of all constituents of the rocket, gases, and body, remains zero, because momentum is a vector quantity and the upward and downward momenta cancel outs.   Thus a rocket does not operate by ‘pushing’ against anything and functions best in the near vacuum of space, where friction is virtually absent.  Energy also is conserved in a rocket, the kinetic energy of the rocket and of the exhaust gases after firing being equal to the chemical energy expended in producing the gases at high velocity.  In review:  rocket propulsion is based on conservation of momentum.  At all times the downward momentum of the exhaust gases is equal to magnitude and opposite in direction to the upward momentum of the rocket itself.