Instantaneous Velocity at a Sharp Point
My friend and I are having an argument. Here's the situation:
A block is at rest on a frictionless surface. At instant $t_0$ a person pushes the block to the right with a constant force. At instant $t_1$ the person stops pushing the block which then glides at a constant velocity $v_f$.
What is the instantaneous velocity of the block at instant $t_1$?
My friend says it's a simple "Newton's Laws" question and the answer is $v_f$ because there are no forces on the block at that time so it stays in constant motion.
I would have agreed with him if the question asked for the instantaneous velocity immediately after $t_1$. But I think the right answer is more obscure than that. There are two ways of thinking about this problem:
1. The Ideal Case
In an ideal case, the force is released instantaneously which would result in the following graphs:
Notably, at $t_1$ the acceleration is discontinuous because the force is released instantaneously. 512
This would mean that the velocity and displacement graphs would have sharp corners at $t_1$. Therefore, in this ideal case, the instantaneous velocity at $t_1$ is undefined because displacement isn't differentiable (the limit from the left is not equal to the limit from the right).
This is quite a generous abstraction from reality that would require an infinite force from your muscles to pull your hand back in an infinitely short amount of time. But I don't think it matters, because even in a realistic case, the instantaneous velocity cannot be $v_f$.
1. The Realistic Case
In reality (a non-ideal case) the force by the hand on the block is not released instantaneously at instant 5. This would mean the acceleration dips down over some period of time as shown below. This would mean the velocity graph would be a curve (differentiable) at $t_1$ rather than a sharp point (undifferentiable). I have drawn the realistic curves in red:
The conservation of momentum tells us that this velocity curve can never exceed $v_f$ so the realistic case is strictly less than the ideal case around $t_1$. In my mind, this is proof that the instantaneous velocity at $t_1$ must be between the instantaneous velocities just before $t_1$ and just after $t_1$.
So who is right? 766