# 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. 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?

• Your position graph is wrong. The velocity gives you the slope of the position, when $v$ becomes constant then you just continue the line of the parabola in $x$ but now as a straight line (you stitch together a parabola with the line tangent to it). – user137661 Oct 30 '19 at 18:28
• You are correct, in the realistic case, there can't be a discontinuity in the acceleration. Hence, it must smoothly go down. So the velocity must be less than the final velocity. – pyroscepter Oct 30 '19 at 18:31
• But also in a realistic case there are no frictionless surfaces and instantaneously applied forces. There is also air resistance. For that matter there are no constant forces either. Even if you release the force gradually the problem is still as idealised as it can get... For an ideal problem the ideal condition seems to be most appropriate. – user137661 Oct 30 '19 at 18:35
• Instantaneous changes of velcoity like that have the same significance as frictionless surfaces; massless pulleys; semi-infinite, rigid rods; infinite square wells; and all the other idealizations of physics teaching. That is, they are good for teaching and good for some kinds of approximating but you must understand them as idealizations. Real system can only approximate instantaneous acceleartion, so the chenge is, in reality, smooth. – dmckee --- ex-moderator kitten Oct 30 '19 at 18:48
• @SV You are right. The graph should continue the line of the parabola in x but as a straight line. My bad, the point was just to differentiate it from a parabolic increase. But you are correct. – AlexanderTheGreat39 Oct 30 '19 at 20:00

Your friend is almost definitely approaching the problem as intended.

The question describes an idealized scenario which specifically calls out a constant force. Your objection revolves around a more realistic, but non-constant applied force.

It also describes the surface as frictionless, and heavily implies that everything happened instantaneously. These are all just simplifications of the problem, because it is trying to draw your attention to a specific concept. The intention isn't to get you to think about all the complex dynamics of how forces get applied over time; the intent seems related to the fact that in this situation, the block will maintain the final velocity when the net force is $$0$$.

It also fits best with the information provided by the question. We don't know the mass, or the applied force. Unless you interpret the question as the force being instantly released at $$v_f$$, there would be no other way to calculate the velocity at time $$t_1$$ without getting more information.

• I understand that these sorts of pedagogical simplifications are necessary. But why is it acceptable to idealize the question ("frictionless surface", "constant force", "no air resistance"), but not the answer? – AlexanderTheGreat39 Oct 30 '19 at 20:12
• @AlexanderTheGreat39 I'm not sure I follow. The answer is also "idealized", specifically because it is constrained by an "ideal" question which doesn't perfectly represent the real world. The "ideal" answer to any question is just the correct one. By constraining the question with idealizations, it means the correct answer is one that conforms to the ideal situation; not the complicated one you described. – JMac Oct 30 '19 at 20:17
• Ah, I see what you are saying now. Thank you for the clarification. – AlexanderTheGreat39 Oct 31 '19 at 0:29

I agree with @JMac answer and the comment of @dmckee (which merits being an answer as well). In addition to all the other real variables mentioned already, we also know that real objects, at least at the macroscopic level, do not behave like ideal rigid bodies. When a force is applied to a non rigid body there will be some amount of deformation, no matter how small, so that the force on the body cannot result in an instantaneous change in acceleration, but rather a more or less gradual change in acceleration. I think it is best illustrated by the red curve around $$t_1$$ in the middle figure of your "Realistic Case".

In my opinion, rather than take into consideration all of he practical limitations in the analysis, it is best to start an analysis based on assumptions of ideal conditions. One can then introduce, preferably one at a time, realistic constraints that are prioritized on the basis of greatest relevance to the analysis at hand.