I’m referring to an instant in time where the velocities of two different particles are identical. One particle experiences a force and the other does not. I think their states of motion are different even if the velocities are identical. Am I correct?
In Figure 1 below, Particle 1 has a velocity to the right and is experiencing a constant force. Particle 2 travels unaccelerated at 10 mps.
In Figure 2 both particles reach points A and B at the same instant in time and the velocities are identical at 10 mps. Particle 1 is experiencing a velocity changing force while Particle 2 experiences no force.
If we try to match the 10 mps velocities, going forward in time, by removing the force on Particle 1 it would have to be at a point past point A which is also a later instant in time. This is because, at the instant in time and location in space of Particle 1 at point A, there is a force on Particle 1. There cannot be both a force and no force on Particle 1 at point A and that instant in time. Thus, if Particle 1 reaches point A and is experiencing the force, the velocities of both particles will never be the same going forward.
When you throw a ball straight up it slows until reaching its apex of travel where for an instant in time there is no velocity. We know it is transitioning from a positive velocity to a negative velocity (relative to earth) but our perception is that there is a pause in the ball’s motion.
If at that instant the ball - with no velocity relative to earth - was experiencing no gravity then it would not move going forward in time. I think that since there is gravitational acceleration at that instant, the ball will move toward earth no matter what at any instant in time later. Math says we can remove the force at that instant and the ball stays put but logic says otherwise.
If we are to believe that the math describes the physics then how could we have, in the two examples, a stationary ball with an applied force and a stationary ball without a force and both having the same results of no velocity at any instant later? This tells me that acceleration is a quantity of motion just as velocity, at any instant in time. That means Particles 1 and 2, though having identical velocities, are not in the same state of motion at points A and B.