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 |
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 mv ^{2} 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 |
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
= F |
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:
J = p _{f} - p _{o} = Δp |
K = |
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.