Mechanics 2: Linear and Rotational Dynamics - 3D Math Primer for Graphics and Game Development - page 584

Force as the Derivative of Momentum

d
dt P (t) = f (t).

(12.21)

The net external force on a system is equal to the rate of change of mo-
mentum of the system.

Equation (12.21) is not just an interesting observation about force and
momentum, it's a completely valid way to define force. In fact, although the
modern presentation of Newton's laws is in terms of forces and masses, when
Newton himself first expressed the laws, he wrote in terms of momentum.
He used the word “motion,” but from his writings we understand that
he used that word in a very particular sense, and he really was talking
about momentum. (The word momentum hadn't been attached to that
concept yet. Remember, he was the guy laying down all the ground rules.)
Newton's second law was originally expressed in a form that more closely
resembles Equation (12.21) than the f = m a form you will more commonly
see.

12.3.1 Conservation of Momentum

Let's return to our investigation into what happens when Moe pushes
against the Earth to get his box moving. Newton's law tells us that the
Earth, not having anything else to push back on, receives a net force, and
thus an acceleration (and a torque, which we discuss later). Yes, you cause
the Earth to accelerate when you push boxes as well as when you take
each and every step! Of course, the Earth's mass is so large compared
to Moe's force that this acceleration is small. Not only that, but Moe's
force pushing the box to the east might be canceled out by Joe's force in
North Dakota pushing his box to the west at the same time. An issue
even more important than these two facts involves the “accounting laws”
of physics: “there is no such thing as free momentum.” Moe doesn't need
Joe to balance out his force; as it turns out, he can't help but do it all by
himself!
Observe that once Moe sets the box in motion, he will need to eventually
stop it. According to Newton's first law, the only way to stop a moving
box is through a force, and according to the third law, this can happen
only if there is some other object involved to receive the opposite force.
Perhaps the box bumps into a tree and comes to a stop. (We consider the
tree to be part of the Earth. Remember that Newton's third law justifies