Momentum is a vector quantity defined as the product of mass and velocity, p = m v. Obviously this makes its direction the same as the direction of the velocity. There is no special unit for momentum, so it is normally expressed in the units kg m / s.

Impulse is a vector quantity defined as the average force acting upon an object multiplied by the time interval through which the force acts, Impulse = FD t. It can also be expressed as the change in momentum or Impulse = Dp. Equating these two definitions of impulse gives FDt = Dp. Dividing both sides of the equation by Dt gives F = Dp / D t which is another form of Newton's Second Law.

In cases in which the net force is zero, this equation tells us that the change in momentum must also be zero. This leads to the conclusion that the total momentum is constant whenever the net force on a system is zero. This is the law of conservation of momentum, which is one of the most useful relationships in physics. If the momentum is known at a particular time, it can be calculated for other times as long as there are no net external forces acting on the objects in the system being considered. We use the conservation of momentum often in analyzing collisions in which we know the momentum before the collision and wish to calculate the velocity of the object after the collision. This use of conservation of momentum allows us to determine the results events such as collisions between objects without having to know the detailed behavior of the forces during the collision. In general it is easier to calculate the momentum before and after a collision than it is to obtain information regarding the complicated force relationships that occur during the collision.

Momentum is a vector quantity, so care must be taken to identify the directions involved. It is easy to make a mistake by not assigning proper direction to the momentum terms in a calculation. For better "bookkeeping" it is helpful to clearly identify the direction you consider as positive whenever you begin a problem and then to carefully reference all subsequent quantities to that direction, affixing negative signs on quantities that point in the opposite direction. As an example, we often choose motion to the right in a horizontal direction as representing the positive x axis. Obviously this choice requires any motion to the left along a horizontal axis to be identified with a negative sign.

Whenever objects stick together after collision, the collision is referred to as a perfectly inelastic collision. Analysis of such events is somewhat simplified, because there is only one velocity to determine after the collision.

Perfectly elastic collisions are those for which the kinetic energy before the collision is equal to the kinetic energy after the collision, that is the kinetic energy and the momentum both are conserved. Recall that Kinetic energy was defined in Chapter 6 as one half the product of the mass times the square of the velocity, K E = 1 /2 m v ².