# Interpreting Aristotle's law of motion $\vec F = R\vec v$

The Aristotle's law of motion, which is incorrect, states that

The velocity of an object $\vec v$ is directly proportional to the force $\vec F$ acting on it

or $\vec F \propto \vec v$

or $\vec F = R\vec v$,

where $R$ is the constant of proportionality which is the resistance to motion.

I want to know how can we interpret this equation $\vec F = R\vec v$ ? Does this equation imply that $\vec F$ and $\vec v$ have the same direction? If it does, then how do we know this through this equation? Does equality mean that they have the same direction?

The book in which I actually saw this equation, wrote this equation in this way:

$$\vec v = \frac {\vec F}{R}.$$

The equation implies that if I push on an object with force $\vec F$, it will start moving at velocity $\vec v = \vec F / R$.

You are correct when you say that having vectors on either side of the equal sign means they will be in the same direction. If I push an object away from me, this equation says it will move in the direction I pushed it. It also says that stronger input forces give faster output velocities. If I just give a small push, the object will move not so quickly, but a big push means fast motion. So far it sounds pretty reasonable, right?

There are several ways this equation doesn't accurately describe motion, though. I will take one illustrative example.

Say an object is sitting in front of me (in space, so there isn't any gravity or friction). I give this object a little push and it starts moving. After my push is done, I'm not imparting any more force to the object. What does this equation say should happen, and what really happens?

This equation says that when my hand is in contact and I am pushing the object, it will move with velocity $\vec v$. But as soon as I take my hand away, the force goes to $0$, therefore the object's velocity goes to $0$ as well. That means the object only moves when I push it and immediately stops dead when I am done pushing.

Of course, we know that in reality if I push an object it will continue moving after the push is done. And in my example, in space with no gravity and no friction, the object will continue moving at a constant velocity forever. That's because in the real world, forces don't cause velocities, forces change velocities; no force, no velocity change.

Presumably $R$ is intended to be a scalar, and multiplying a vector by a scalar just changes the magnitude of the vector without changing it's direction. So yes, $\vec{F}$ and $\vec{v}$ would have the same direction.

In the real world $R$ might be a tensor, and in that case $\vec{F}$ and $\vec{v}$ are not guaranteed to point in the same direction. This isn't as complicated as it sounds. Suppose you push a railway train at an angle to the direction of the rails. The train will move along the rails, which is a different direction to the one you're pushing in. This is a somewhat contrived example, but you get similar affects in materials that are anisotropic.

There is a sense wherein Aristotle was correct (at least in Newtonian physics). Namely, if $\vec{F}_{aristotle}$ is the vector quantity we call impulse that measures the total "shove" you give an object to set it going. It's quite intuitive: the harder you shove a shopping trolley, the swiftlier it runs away from you.

In Newton's terms:

$$\vec{F}_{aristotle} = \int_{t_0}^{t_1} \vec{F}_{newton} \,{\rm d} t = m \vec{v}\qquad(1)$$

and then the quantity $R$ in your equation is what we call the object's inertia or mass: $\vec{F}_{aristotle} = m\,\vec{v}$, where $\vec{v}$ is the vector velocity of a body, which is stationary relative to us to begin with, after we have given it a "shove" $\vec{F}_{aristotle}$.

We don't know exactly what Aristotle had in mind to be the force. Where we know he was wrong was that he said that his concept of the force had to be applied continuously, without letting up, to keep a body moving, but this mistake was not needfuly embodied in his $\vec{F} = R \vec{v}$ law. Newton's insight that corrected Aristotle was to understand that bodies without interaction with the outside World do not change their state of motion. Aristotle overlooked that, once you get a body going with a "shove", things like air resistance and friction have their own "shove" back and that, with most everyday things we don't often see objects that truly do not interact with the outside World. Once you have this understanding, i.e. the grasping of Newton's first law, then it makes much more sense to define "interaction" as the time rate of change of the acted upon body i.e. it makes sense to focus on acceleration as measuring the instantaneous interaction as opposed to the velocity change which measures the "smeared out" interaction over time. Especially so when you were the one who invented differential calculus and had derived the fundamental theorem of calculus. So Newton simply took the time derivative of the Aristotlean version (1):

$$\vec{F}_{newton} = {\rm d}_t(m\,\vec{v})\qquad(2)$$

I'd argue that Newton's first law is the crucial insight and its negation is what defines interaction; once you gasp this, then the second law becomes really a definition of nett force on a body. As I have argued, the second law is not needfully incompatible with Aristotle's (1) (if indeed he did mean impulse); it is the first law that Aristotle got wrong. See for instance my answer to "Explaining Newton's Laws of motion to a 6 year old".