Having studied the macroscopic movement of a system of particles, we now turn to the microscopic movement: the movement of individual particles in the system. This movement is determined by forces applied to each particle by the other particles. We shall examine how these forces change the motion of the particles, and generate our second great law of conservation, the conservation of linear 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:
|J = FΔt|
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:
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:
|p = mv|
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.
Equation 1: Relating Force and Acceleration
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:
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.