Newton’s second law definition

Newton's Second Law as stated below applies to a wide range of physical phenomena, but it is not a fundamental principle like the Conservation Laws. It is applicable only if the force is the net external force. It does not apply directly to situations where the mass is changing, either from loss or gain of material, or because the object is traveling close to the speed of light where relativistic effects must be included. It does not apply directly on the very small scale of the atom where quantum mechanics must be used.

Differential form:      Force =Change of momentum with change of time     
With mass constant:  Force = mass  acceleration                                          F = ma

Force, acceleration, momentum and velocity are all vector quantities.

Each has both a magnitude and a direction

Sir Isaac Newton first presented his three laws of motion in the "Principia Mathematica Philosophiae Naturalis" in 1686. His second law defines a force to be equal to the differential change in momentum per unit time as described by the calculus of mathematics, which Newton also developed. The momentum is defined to be the mass of an object m times its velocity v. So the differential equation for force F is:
F = d(m * v) / dt
If the mass is a constant and using the definition of acceleration a as the change in velocity with time, the second law reduces to the more familiar product of a mass and an acceleration:
F = m * a
Since acceleration is a change in velocity with a change in time t, we can also write this equation in the third form shown on the slide:
F = m * (v1 - v0) / (t1 - t0)
The important fact is that a force will cause a change in velocity; and likewise, a change in velocity will generate a force. The equation works both ways. The velocity, force, acceleration, and momentum have both a magnitude and a direction associated with them. Scientists and mathematicians call this a vector quantity. The equations shown here are actually vector equations and can be applied in each of the component directions. The motion of an aircraft resulting from aerodynamic forces and the aircraft weight and thrust can be computed by using the second law of motion.

Newton's Second Law Illustration

Newton's 2nd Law enables us to compare the results of the same force exerted on objects of different mass.

Limitations on Newton's 2nd Law

One of the best known relationships in physics is Newton's 2nd Law

but, though extremely useful, it is not a fundamental principle like the conservation laws. F must be the net external force, and even then a more fundamental relationship is

The net force should be defined as the rate of change of momentum; this becomes

Only if the mass is constant. Since the mass changes as the speed approaches the speed of light, F=ma is seen to be strictly a non-relativistic relationship which applies to the acceleration of constant mass objects. Despite these limitations, it is extremely useful for the prediction of motion under these constraints.