Momentum is a derived quantity, calculated by multiplying the mass, *m* (a scalar quantity) times velocity, ** v** (a vector quantity). This means that the momentum has a direction and that direction is always the same direction as the velocity of an object's motion. The variable used to represent momentum is

**. The equation to calculate momentum is shown below.**

*p*Equation for Momentum:=

pmv

The SI units of momentum are kilograms * meters per second, or kg*m/s.

### Vector Components and Momentum

As a vector quantity, momentum can be broken down into component vectors. When you are looking at a situation on a 3-dimensional coordinate grid with directions labeled *x*, *y*, and *z*, for example, you can talk about the component of momentum that goes in each of these three directions:

p=_{x}mv=_{x}

p_{y}mv=_{y}

p_{z}mv_{z}

These component vectors can then be re-constituted together using the techniques of vector mathematics, which includes a basic understanding of trigonometry. Without going into the trig specifics, the basic vector equations are shown below:

=p+p_{x}+p_{y}=p_{z}mv_{x}+mv_{y}+mv_{z}

### Conservation of Momentum

One of the important properties of momentum and the reason it's so important in doing physics is that it is a *conserved* quantity. That is to say that the total momentum of a system will always stay the same, no matter what changes the system goes through (as long as new momentum-carrying objects are not introduced, that is).

The reason that this is so important is that it allows physicists to make measurements of the system before and after the system's change and make conclusions about it without having to actually know every specific detail of the collision itself.

Consider a classic example of two billiard balls colliding together. (This type of collision is called an *elastic collision*.) One might think that to figure out what's going to happen after the collision, a physicist will have to carefully study the specific events that take place during the collision. This actually isn't the case. Instead, you can calculate the momentum of the two balls before the collision (** p_{1i}** and

**, where the**

*p*_{2i}*i*stands for "initial"). The sum of these is the total momentum of the system (let's call it

**, where "T" stands for "total) and after the collision—the total momentum will be equal to this, and vice versa. (The momenta of the two balls after the collision is**

*p*_{T}**and**

*p*_{1f}**, where the**

*p*_{1f}*f*stands for "final.") This results in the equation:

Equation for Elastic Collision:=

p_{T}+p_{1i}=p_{2i}+p_{1f}p_{1f}

If you know some of these momentum vectors, you can use those to calculate the missing values, and construct the situation. In a basic example, if you know that ball 1 was at rest (** p_{1i}** =

**0**) and you measure the velocities of the balls after the collision and use that to calculate their momentum vectors,

**and**

*p*_{1f}**, you can use these three values to determine exactly the momentum**

*p*_{2f}**must have been. (You can also use this to determine the velocity of the second ball prior to the collision, since**

*p*_{2i}**/**

*p**m*=

**.)**

*v*Another type of collision is called an *inelastic collision*, and these are characterized by the fact that kinetic energy is lost during the collision (usually in the form of heat and sound). In these collisions, however, momentum *is* conserved, so the total momentum after the collision equals the total momentum, just as in an elastic collision:

Equation for Inelastic Collision:=

p_{T}+p_{1i}=p_{2i}+p_{1f}p_{1f}

When the collision results in the two objects "sticking" together, it is called a *perfectly inelastic collision*, because the maximum amount of kinetic energy has been lost. A classic example of this is firing a bullet into a block of wood. The bullet stops in the wood and the two objects that were moving now become a single object. The resulting equation is:

Equation for a Perfectly Inelastic Collision:

m_{1}+v_{1i}m_{2}v_{2i}= (m_{1}+m_{2})v_{f}

Like with the earlier collisions, this modified equation allows you to use some of these quantities to calculate the other ones. You can, therefore, shoot the block of wood, measure the velocity at which it moves when being shot, and then calculate the momentum (and therefore velocity) at which the bullet was moving prior to the collision.

### Momentum and the Second Law of Motion

Newton's Second Law of Motion tells us that the sum of all forces (we'll call this ** F_{sum}**, though the usual notation involves the Greek letter sigma) acting on an object equal the mass times acceleration of the object. Acceleration is the rate of change of velocity. This is the derivative of velocity with respect to time, or

*d*/

**v***dt*, in calculus terms. Using some basic calculus, we get:

=F_{sum}m=am*d/vdt=d(m)/vdt=d/pdt

In other words, the sum of the forces acting on an object is the derivative of the momentum with respect to time. Together with the conservation laws described earlier, this provides a powerful tool for calculating the forces acting on a system.

In fact, you can use the above equation to derive the conservation laws discussed earlier. In a closed system, the total forces acting on the system will be zero (** F_{sum}** =

**0**), and that means that

*d*

**P**_{sum}/

*dt*=

**0**. In other words, the total of all momentum within the system will not change over time which means that the total momentum

*P*_{sum}*must*remain constant. That's the conservation of momentum!