Thank you all for the insightful comments on my previous question Are the concepts of motion such as mass, momentum, impulse, work, energy, force etc. fictitious/abstract concepts or are they real "things"?, but I may have misspoken and there has been a bit of misunderstanding which is entirely my fault. I've ignorantly interchanged "real" with "physical significance". What I meant to ask is what are these concepts' physical significance? For example, momentum, which I have been taught to define as the product of the mass and velocity of an object; what is its physical significance? If a metre is a measurement of distance then what does momentum measure? Same question applies to energy, etc.
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Momentum is a measure of an object's resistance to any force that might slow it down, stop it, or in any way change its state of motion (speed and/or direction). The greater an object's change of momentum, the greater the force, more specifically impulse (which is numerically equal to the force multiplied by time), that is required to make this change.
As for the physical significance of the energy of an object, this depends on the type of energy we are talking about. But in general, you can think of it as the amount of work you need to do by a force to the object to bring about this energy, numerically equal to the force multiplied by distance.
In fact, you may have already learned that $$\text{force}\times\text{time} =\text {change in Momentum=impulse}$$ and $$\text{force}\times\text{distance}=\text{change in Energy = work done}$$
It's also important to note that both momentum and energy are conserved quantities, which makes them powerful tools when determining the dynamics of systems and how systems evolve over time.
joseph h
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Velocity can be defined as the momentum of an object divided by its mass. What is its physical significance?
Of course, we can give both momentum and velocity (and mass) definitions that don't (directly) depend on each other. For example, momentum is proportional to the force needed to bring the object to a stop in a fixed amount of time, velocity is proportional to the distance the object travels in a fixed amount of time, and mass is proportional to the force needed to change the object's velocity by a fixed amount in a certain time. But (at least in classical Newtonian physics) none of these definitions is really more fundamental than the others.
In particular, in Newtonian physics, if we know any two of an object's momentum, velocity and mass, we can calculate the third quantity. It seems like you're used to treating velocity and mass as "fundamental" properties of objects, and momentum as something calculated from those, but that's a completely arbitrary choice. You could just as well treat e.g. momentum and mass as fundamental properties, and velocity as something calculated, if needed, by taking their ratio.
Indeed, in Einstein's theory of relativity, it turns out that massless particles such as photons all have the same absolute velocity (the speed of light) and mass (zero) even though their momentum can be different (in magnitude as well as direction)! Thus, in relativistic physics, specifying the mass and velocity of a particle is not sufficient to determine its momentum. Knowing the momentum and mass of an object is, however, enough to determine its velocity. In that sense, in relativistic physics, momentum is more fundamental than velocity.
Also, even in classical physics, while the total momentum and total mass of a closed system are conserved quantities, total velocity is not: it can change e.g. when momentum is transferred from a heavy object to a light one or vice versa. That's another reason for treating velocity as less fundamental than momentum even in classical physics.
Indeed, in principle Newtonian physics may be formulated entirely without the concept of velocity, just by substituting "momentum / mass" for "velocity" everywhere (and simplifying the resulting formulas where appropriate). In some cases such as reformulation will even be simpler than the "traditional" formulation using acceleration: for example, instead of first defining acceleration as the change in velocity over time, and then force as mass × acceleration, we can just define force directly as the change in momentum over time.
That said, of course, the concept of velocity isn't completely useless, either. In particular, in many practical cases the velocity of a moving object is much easier to measure accurately than either its momentum or mass.
Also, working with velocities and accelerations (as opposed to momenta and forces) can be useful when working with objects that (only) experience fictitious forces proportional to their mass, since treating these "fictitious forces" as "anomalous accelerations" instead allows the trajectories of such objects to be determined without knowing their mass.
This also works with gravity, which also exerts a force on an object proportional to its mass (and thus an acceleration independent of its mass!). In fact, in general relativity gravity is treated as a fictitious force arising from the curvature of spacetime. Even in Newtonian physics, working with velocities and accelerations makes it easier to e.g. calculate the trajectories of objects of negligible mass in a gravitational field, such as satellites in Earth orbit.
Ilmari Karonen
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