Newton's Second Law of Motion

Newton's 2nd law is really a mathematical statement of how "force" relates to the change in an object's motion. In part, it acts as a defining statement for the properties termed "force" and "mass."

We already know that mass is a measure of the degree of inertia an object possesses, and we intuitively understand what force is - but we don't really have independent definitions of them or procedures for measuring them. The second law relates them to each other, and in so doing, acts as a joint operational definition of them.

Mathematically, Newton's 2nd law says F = Δp/Δt. In words, we might say that force, F, is equal to the change in momentum, Δp, with the corresponding change in time, Δt. This brief mathematical statement contains quite a bit of information.

First, what is momentum?

By definition, p = mv, where p is the property called momentum, m is the object's mass, and v is its velocity. [In this equation, and above, you see that some of the letters in the equation are in bold-italic while others are just italic. The items designated in bold-italic are called vectors; they have mathematical properties of size and direction. The items in italic are scalars - regular numbers denoting size only. So mass, m, is a scalar defined only by its size while velocity, v, is a vector designating the size of an object's speed and which direction it is traveling in.]

Newton recognized that the change in an object's motion with an applied force depended also on its mass (degree of inertia). This is a common sense idea to us; if we push with a certain amount of force on a toy car and with the same amount of force on a real car, the toy car will have a much greater change in its motion. Momentum incorporates inertia (mass) with an object's motion (speed) to produce a composite that relates to force.

Accelerate

Decelerate

Sideways

The relationship with force is to the rate at which momentum changes over time! [The Δ symbol in the equation means "change in."] So, force F produces a change in an object's momentum p over time, where the size of the force defines the rate of change and the direction of the force defines the direction of the change. If the force's direction lines up with the object's original direction of travel, the object's speed will increase (force in the same direction as the object's velocity) or decrease (force in the opposite direction to the object's velocity). But if the force points in some other direction, it will cause the object to change course as well.

As long as the mass part of the momentum does not change, the change in momentum can be expressed as Δp = Δ(mv) = m(Δv). Because we define acceleration, a, as the change in velocity with time a = Δv/Δt, Newton's second law can also be written as F = ma. This is a more familiar expression to anyone who had a physics course before. [It is interesting to note that since the acceleration a relates to Δv, and Δv is not zero if the direction of v changes even when its size stays the same, an object accelerates if its direction of travel changes while it is moving at constant speed.]

Newtons 2^nd law can also be understood to describe the action of many forces acting on a single object, or many forces acting on many objects. Under these circumstances, the force F is understood to represent the overall, or net, force and the momentum change Δp to be the overall, or net change - over all of the objects involved.