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:
$$\vec v = \frac {\vec F}{R}. $$
Best Answer
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.