Is it possible to have a logically consistent world where F=mv instead of F=ma?
And could it look anything like the universe we know?
Don't get hung up on the terminology. In general, you seem to see through it easily, so I don't want any more answers mentioning that. I'll say it's a translation error: My "F" is the quantity that, in the equations of motion discovered by physicists in this universe, occupies the position most similar to that occupied by force in our universe. Things - I guess I can't call them "forces" anymore - like gravity and magnetism confer a velocity rather than an acceleration.
I ask because this way is ancient logic, "common sense", the easier way for humans to understand. But if you look more at how such a world would function, what's impossible? I was thinking about Aristotle's "natural motion" - though don't try to be consistent with Aristotle in your answers. He believed there could be no vacuum because the only thing resisting motion was fluid resistance - that is, inertia was 0 - and any applied force to an object in vacuum would result in infinite speed. I'm taking a different approach, as explained next.
A world with no momentum. If you stop pushing a rolling object, it stops. Objects fall at a constant speed in a vacuum. I may have meant to say "no inertia" here. What I mean is that, where in our universe inertia resists acceleration, in this universe the equivalent to inertia would resist motion.
A recent answer required this clarification: I recognize that a universe entirely filled with highly viscous fluid would create a somewhat similar situation using physics as we know them. Objects would require force to remain in motion. However, I am not looking for that answer.
With real physics, if you are moving and encounter a viscous medium, you will slow down, and you will feel the deceleration. In my world, you feel speed, not acceleration. Driving into a wall won't kill you, but driving too fast will. There's also the minor matter that fluid resistance is, I believe, roughly proportional to speed squared. In my universe, the (force-like quantity) required to move in a vacuum is defined as linearly proportional to speed.
Orbits are impossible. What could the astronomical-scale universe look like?
Are any form of atoms even possible, or would matter have to be continuous?
Are fluids possible? How might they behave? This seems to be the most critical factor in creating a recognizable world.
Assume some form of weak anthropic principle. That's the question: What other laws have to be different to create a universe that could still contain something we could recognize as intelligent life capable of drawing the conclusion (force-like-quantity)=mv?
This post was sourced from https://worldbuilding.stackexchange.com/q/4737. It is licensed under CC BY-SA 3.0.
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tl;dr: Such a world would be quite different from ours. Basically, all modern formulations of classical mechanics fail on it, and you also could not base that world on an underlying quantum theory.
I assume that you want the laws of physics as similar to ours as possible with this restraint. Especially I assume that you want to preserve an underlying quantum theory for which the macroscopic world is the classical limit.
This implies that the world would be governed by a principle of extremal action (the action being essentially the phase of the quantum wave function), and therefore the classical world being described by Lagrangian equations with an appropriate Lagrangian.
Since the intended law of motion does not include acceleration, the Lagrangian must be linear in the velocity, as otherwise the Lagrange equations would generate an acceleration term. So the Lagrangian must have the form $$L(\vec x,\dot{\vec x}) = \vec f(\vec x)\cdot\dot{\vec x} + g(\vec x)$$ Inserting into the Lagrangian equations $$\frac{\mathrm d}{\mathrm dt}\frac{\partial L}{\partial\dot x_k} = \frac{\partial L}{\partial x_k}$$ we get $$\vec\nabla f_k(\vec x)\cdot\dot{\vec x} = \frac{\partial \vec f(\vec x)}{\partial x_k}\cdot \vec x + \vec\nabla g(x)$$ Now the meaning of $g(x)$ is clear: It's just the negative of a "potential function" (quotes, because it is not really a potential function, just like your force is not really a force, as the units show). So let's write $g(x)=-V(x)$. WE have then $\vec F=-\vec\nabla V$ just as in conventional mechanics.
For $\vec f(\vec x)$ the situation is a bit more complicated: To get the intended equation of motion, we need $$\vec\nabla f_k(\vec x)-\frac{\partial \vec f(\vec x)}{\partial x_k} = m\vec e_k$$ for all $k$. However, let's write this down for the first component of $k=1$: $$\frac{\partial f_1(\vec x)}{\partial x_1} - \frac{\partial f_1(\vec x)}{\partial x_1} = m$$ Clearly for $m\ne 0$ this equation cannot be fulfilled, since the left hand side is identically $0$. In other words, your law of motion cannot be derived from an action principle, which means that your world cannot be based on quantum mechanics. Given that quantum mechanics in our world is responsible for a lot of the behaviour of materials (including the very fact that there are solid materials to begin with), this means your world must be very different from ours.
Also note that all modern formulations of classical physics (Lagrange, Hamilton, Hamilton-Jacoby) ultimately depend on the stationary action principle. So basically the complete tool set of modern physics could not be applied to your world.
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