Newton’s Second Law of Motion (the law of “force and acceleration”) states:

“The change of motion [acceleration] is proportional to the motive force impressed; and is made in the direction of the right [straight] line in which that force is impressed.” [1]

Unlike inertial motion, this law describes the motion of a body which is not in equilibrium, vis., where a force is acting on the body and is not counterbalanced by another force. Newton defined “impressed force” as “an action exerted upon a body, in order to change its state…” [2]

Newton intended the term “change of motion” to mean “change in velocity” or, in modern terminology, “acceleration.”[3] Newton’s second law, in effect, states that the acceleration (a) of a mass (m) is in the direction of the continuous net force (F) applied to it and is proportional to such force[4] (Figure 9). Therefore, in empty or free space: F = ma.[5]

Newton defined the “mass” (m) of an object to mean its “quantity of matter…its density and bulk [volume] conjointly.”[6] The greater a body’s mass, the more a body ”resists” being accelerated (having its state or direction of motion changed).

Empirically, it is observed that: “If a force causes a large acceleration, the mass of the [accelerated] body is small; if the same force causes only a small acceleration, the mass of the [accelerated] body is large”[7] (Figure 9). Thus, very importantly, the mass of a body is the quantitative measure of its power or force of inertial resistance, and the magnitude of mass is inversely proportional to the acceleration caused by a continuously applied force: m = F/a.[8] Since “mass” is the quantitative measure of matter’s inertial power of resistance, the magnitude of a body’s mass is often referred to as “inertial mass.” On Earth, inertial mass (m) is more correctly described as m=F/(a-R).

Reciprocally, and very importantly, Newton’s second law also asserts that the magnitude of acceleration of a body is also inversely proportional to the mass of the accelerated body.[9] Feynman agreed: “[A] body reacts to a force by accelerating, or by changing its velocity every second to an extent inversely to its mass.”[10] Thus, in free space a = F/m, or on Earth (a – R) = F/m.

How do we know that this formula is correct? Because empirically if you apply the same continuous force to two barrels (m1 and m2) and one barrel (m1) accelerates only one-fifth as fast as the other barrel (m2), then the slower barrel (m1) must have 5 times as much mass as the faster barrel (m2), and the magnitude of acceleration (a) of each barrel is inversely proportional to the magnitude of its own mass.[11] Also, in this terrestrial example, each barrel with a different mass receives the same quantity of applied force.[12]

The above terrestrial example can also be described in another way. The greater (motion) magnitude of acceleration (a2) of the lesser mass (m2) is proportional and equivalent to the greater inertial resistance of the greater mass (m1)…and the lesser (motion) magnitude of acceleration (a1) of the greater mass (m1) is proportional and equivalent to the lesser inertial resistance of the lesser mass (m2). Thus, “the ratio of masses [their inertial resistances] is the inverse of the ratio of the accelerations [their motions]:” m2/m1 = a1/a2.[13] These concepts and ratios become critical when we discuss relative gravitational accelerations: of opposing gravitating bodies in later sections of this treatise[14] (bottom of Figure 9).

At the beginning of the Principia, Newton defined the “quantity of motion [as] the measure of the…velocity and quantity of matter conjointly.”[15] In other words, the term “quantity of motion” (as defined by Newton) means the “mass (m) times the velocity (v)” of a body. Since the time of Newton this concept has been referred to as “momentum,” or p.[16] Thus p = mv. The more mass and/or velocity that an object has, the more momentum it has, and therefore the more force which must be applied to slow it down or change its direction of motion.[17]