Conservation of Momentum

Often in systems of particles, two particles interact by applying a force to each other over a finite period of time, as in a collision. The physics of collisions will be further examined in the next SparkNote as an extension of our conservation law, but for now we will look at the general case of forces acting over a period of time. We shall define this concept, force applied over a time period, as impulse. Impulse can be defined mathematically, and is denoted by J:

Just as work was a force over a distance, impulse is force over a time. Work applied mostly to forces that would be considered external in a system of particles: gravity, spring force, friction. Impulse, however, applies mostly to interactions finite in time, best seen in particle interactions. A good example of impulse is the action of hitting a ball with a bat. Though the contact may seem instantaneous, there actually is a short period of time in which the bat exerts a force on the ball. The impulse in this situation is the average force exerted by the bat multiplied by the time the bat and ball were in contact. It is also important to note that impulse is a vector quantity, pointing in the same direction as the force applied.

Given the situation of hitting a ball, can we predict the resultant motion of the ball? Let us analyze our equation for impulse more closely, and convert it to a kinematic expression. We first substitute F = ma into our equation:

But the acceleration can also be expressed as a = . Thus:

The large impulse applied by the bat actually reverses the direction of the ball, causing a large change in velocity.

Recall that when finding that work caused a change in the quantity mv² we defined this as kinetic energy. Similarly, we define momentum according to our equation for an impulse.

From our equation relating impulse and velocity, it is logical to define the momentum of a single particle, denoted by the vector p, as such:

Again, momentum is a vector quantity, pointing in the direction of the velocity of the object. From this definition we can generate two every important equations, the first relating force and acceleration, the second relating impulse and momentum.

The first equation, involving calculus, reverts back to Newton's Laws. If we take a time derivative of our momentum expression we get the following equation:

Thus

It is this equation, not F = ma that Newton originally used to relate force and acceleration. Though in classical mechanics the two equations are equivalent, one finds in relativity that only the equation involving momentum is valid, as mass becomes a variable quantity. Though this equation is not essential for classical mechanics, it becomes quite useful in higher-level physics.

The second equation we can generate from our definition of momentum comes from our equations for impulse. Recall that:

Substituting our expression for momentum, we find that:

This equation is known as the Impulse-Momentum Theorem. Stated verbally, an impulse given to a particle causes a change in momentum of that particle. Keeping this equation in mind, momentum is conceptually quite similar to kinetic energy. Both quantities are defined based on concepts dealing with force: kinetic energy is defined by work, and momentum is defined by impulse. Just as a net work causes a change in kinetic energy, a net impulse causes a change momentum. In addition, both are related to velocity in some way. In fact, combining the two equations K = mv² and p = mv we can see that:

This simple equation can be quite convenient for relating the two different concepts.

This section, dealing exclusively with the momentum of a single particle, might seem out of place after a section on systems of particles. However, when we combine the definition of momentum with our knowledge of systems of particles, we can generate a powerful conservation law: the conservation of momentum.