What happens when a car starts moving? The last moment the car is at rest versus the first moment the car moves

Question

Imagine a car that's at rest and then it starts moving. Consider these two moments:

1. The last moment the car is at rest.
2. The first moment the car moves.

The question is: what happens between these 2 moments? It might sound like a silly question and it probably is: my physics teacher asked us this question years ago and never gave us the answer. Apparently the answer "the 2 moments coincide" was not a valid answer.

I recently learned that he passed away so I will never know if he was just teasing us or if it was a reasonable question. Do you know what the answer might be?

Answer 1

Suppose the last moment the car was at rest is at $t_0=0$. What time is the first moment the car moves?

Suppose you pick some number $t_{first} > 0$. It doesn't matter what number you pick. You can always show that there is a problem with it.

There is a number $t_{first}/2$. This number is bigger than $0$, so it is after $t_0$. So the car is not at rest. It is before $t_{first}$, so the car has not started moving yet.

This is a contradiction. It shows you cannot pick a moment after $t_0$ that is the moment the car first moves. You must conclude the car starts moving at $t_0$. So you are right to be suspicious of his answer.

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This question is continuing to generate interest, so I thought I would add to my answer.

First numbers make a difference. This paradox was first proposed by Zeno. It wasn't settled for thousands of years. But it is quite straightforward to modern mathematicians.

That said, you still have to be careful with it. I haven't been careful enough. The resolution I proposed is that at $t = t_0$, the car is both at rest and has started moving. This sounds like a paradox.

That's OK - I'm a physicist. The purpose of physics is to model the behavior of the universe. A logical inconsistency this small doesn't change the predicted behavior of the car at all. Physicists make bigger approximations all the time.

But it isn't good enough for a mathematician. Mathematicians model ideas. Logic is their only tool. If the logic isn't airtight, a false statement can be proven. And one false statement can be used to prove more. The entire structure of mathematics would collapse.

So a mathematician would pay more careful attention to definitions. What exactly is meant by "at rest", or "has started moving"? Is the car at rest if its position hasn't changed? If its velocity is $0$?

He would note that I haven't really proven that the car has started moving at $t_0$. Just at all times after $t_0$. There is no such thing as the first moment where the car's position has changed. There is only a last moment where it has not.

This kind of distinction is crucial to point set topology. It determines many interesting properties of topological spaces. (Where but CalTech would an introduction like the link come from the Division of Humanities and Social Studies?)

Answer 2

This is a version of Zeno's arrow paradox. One solution is to reject the implied dichotomy of "at rest" or "moving" and say that there is a time $t$ such that for all times before $t$ the car is at rest and for all times after $t$ the car is moving but that at the moment $t$ itself the car is neither at rest nor moving. There is therefore no last moment when the car is at rest, and no first moment when it is moving.

This like saying that there is no largest negative real number and no smallest positive real number because in between the negative and the positive numbers there is a number zero that is neither negative nor positive.

Of course, there is no single "right" answer to this type of question - your teacher's goal when asking the question was to get you to think about the nature of time and the nature of movement, and he has clearly succeeded since since you still remember the question.

Answer 3

Your answer is entirely reasonable - the two moments coincide. There is no physical sense of "between" two time points that have arbitrarily little duration between them. There is no point in time where the car has ceased being at rest, but has not yet started moving. There are only two mutually exclusive states, the car cannot be both not at rest and unmoving.

"Between" is the wrong word to use. We can talk about what happens "at" the moment the car stops being at rest and start moving, since that is all one moment. At that moment, quite simply, the car starts moving.

To put it another way, consider that two values separated by an infinitesimal difference are actually exactly equal to another (as described here). For any duration you could suppose between when the car stops resting and when it starts moving, we can observe there is no such transition that takes that long - the duration between the two time points is infinitesimally small, therefore, the two time points are the same. They're just different ways of naming the same time point - it's a bit like asking what happens between 12:00PM and noon. The concept of "between" requires two distinct objects and is semantic nonsense when applied to only one object. The aforementioned question is more clearly nonsensical when rephrased as "what happens between noon and noon?".

Answer 4

The two moments you mention are not special. You can ask the same question about any other pair of speeds that are infinitesimally close. For example, you can consider the last moment when the car wasn't going more than ten miles an hour and the first moment that it was. Given that, it is wrong to consider them to be the same moment. To see why, consider a series of infinitesimally close speeds, and label them S0, S1, S2, S3 and so on. If you consider that the car is going at S0 then S1 'in the same moment', then you have to apply the same argument to S1 and S2, then to S2 and S3 and so on, which leads you to the obviously false conclusion that the car is going at all speeds at once. If you want a proper understanding of the principles involved here, you are really talking about the fundamentals of calculus, since that's how you are modelling the acceleration of the car.

Of course, all of what I have just said is a huge simplification anyway. If you really were to zoom-in in terms of time to examine the motion of the car over a phenomenally short timescale, you would find that it was never completely still to begin with. The constituent atoms and molecules of the car would all be vibrating randomly with heat energy, and as the car did start to accelerate it would not do so monolithically- instead it would behave in an hugely complicated way, with the movement of some parts lagging behind that of others.

Answer 5

There's a fundamental problem with this question that makes it unanswerable in real physical terms. To understand why, we need to think about physics for a moment and how mathematical models are not reality (the map is not the territory.)

Let's start with some basic concepts. Movement is a change of location. Movement inherently requires the passage of time. Speed is (in this context) the rate of movement over time. So, when we say something is moving or has moved, what we mean is that at time A, an object was at a certain location and at time B, it was at some other location. So, another way to ask your question is "at what time did the speed of the car go from 0 to something more than 0?"

It's fine to come up with logical arguments about this and there's a long history of that. But physics is not just logical arguments and math problems. What distinguishes physics (and science in general) is empirical observations.

How can we experimentally determine when this movement starts? One option would be to set up a video camera with a known frame rate. We can examine the frames and at find the first one where the vehicle's position has changed. Great. But there's a problem: we only know which interval the movement started. That is, we know that movement started between when that frame was taken and when the prior frame was taken. But during that interval, we have the exact same problem as we started with. So, we get a high-speed camera and shorten the interval. We get an even faster camera. We use a laser instead. We keep working to make the interval smaller. Ultimately, though, we hit the limits of precision of our technology. No matter how precise the technology is, though, we will still only identify an interval, not a moment.

And mainstream physics says this isn't just a limit of our current technology, but rather a fundamental aspect of nature. There's simply no way to take a measurement without some finite interval of time. The best we will ever be able to say is that at time A it was at position X and at time B it was at position Y. What happens during that smallest measurable interval of time is a matter of conjecture.

Answer 6

You have entered a rabbit hole that kept physicists and philosophers arguing for ages.

The desciption you gave of a car at rest is a classical description in terms of Newton's mechanics. If we assume continuous values for position and velocity, the question of "between moments" becomes ill-defined, since there is no positive real number "next to 0", as others pointed out.

Quantum mechanics, however, gives a different description, where particles do not have position and momentum vectors of infinite precision. Even the lowest energy state is not exactly zero, some positive amount of zeropoint energy remains. We also can't cool particles down to 0 Kelvin and stop them for the same reason. In essence, nature solves this conundrum by making sure nothing is ever at rest. Everything is vibrating all the time.

Respect to your professor for giving young minds a tough nut to crack. Respect to you for still pondering it.

Answer 7

Imagine a car that's at rest and then it starts moving. Consider these two moments:

The last moment the car is at rest.

The first moment the car moves.

The question is: what happens between these 2 moments?

Note, this is a math puzzle, not a physics puzzle.

Apparently the answer "the 2 moments coincide" was not a valid answer

That is correct. "The 2 moments coincide" is not a valid answer. The key error is assuming that both of those moments exist in the first place. Because the question assumes both exist, it is a trick question. It is possible for one or the other to exist, or neither, but not both.

At some moment, $t$, "the car is at rest" means $v(t)=0$ and "the car moves" means $v(t)\ne 0$ where $$v(t)=x'(t)=\lim_{h\rightarrow 0}\frac{x(t+h)-x(t)}{h}$$ So then "the last moment the car is at rest" is $\max\{t|v(t)=0\}$ and "the first moment the car moves" is $\min\{t|v(t)\ne 0\}$

In order for this limit to exist the position $x(t)$ must be differentiable. Not all functions are. If $x(t)$ is not differentiable then there is no max or min $t$ as described above.

For example, if we imagine (since this is a math exercise rather than a physics exercise) that $$x(t)=\begin{cases} t & 0\leq t \\ 0 & t<0 \end{cases}$$For this function, the above limit does not exist at $t=0$ so $v(0)$ is undefined, meaning for $t=0$ the car cannot be said to be moving, nor can it be said to be at rest. The car is not moving for any negative $t$, but there is no maximum negative number so there is no last moment that it is at rest. The car is moving for any positive $t$, but there is no minimum positive number so there is no first moment that it is moving.

Imagine instead that $$x(t)=\begin{cases} t^2 & 0\leq t \\ 0 & t<0 \end{cases}$$ this function is differentiable at $t=0$ with $v(0)=0$, meaning the car is not moving for any non-positive $t$, and the car is moving for any positive $t$. The maximum non-positive number is $0$ so the last moment the car is at rest is $t=0$. There is no minimum positive number, so there is no first moment that the car is moving.

At any $t$ such that $x(t)$ is differentiable, then $v(t)$ will have one and only one value. That value will either be zero or it will be non-zero, but not both. So any given $t$ cannot both be the first moment that it moves and the last moment that it does not move. Therefore, "the 2 moments coincide" is indeed an incorrect answer to a trick question.

Answer 8

If your teacher does not accept "the two moments coincide" as an answer, then maybe your teacher believed in infinitesimals. Someone made sense of non-standard analysis, but it is not usually used in mathematics.

Answer 9

The car, like almost everything else, is always moving. Or all the bits of it are anyway. When it starts to get going all in one particular direction, some of it will start to get going before the rest of it. So it probably does make sense to talk of a period between the last point when none of it was going in any particular direction (aka 'not moving' when considered on a grander scale), and when all of it was first going in one particular direction. The same would apply to a golf ball being hit. The side towards the golf club gets moving first, deforming the ball, whilst the side away from the club remains more or less stationary for a very small amount of time.

Answer 10

This is essentially a TL;DR of Dale's version:

If the velocity $v(t)$ is a continuous function in the time $t$ (which is plausible for a car with limited accelearation), then $v^{-1}(0)$ is a closed set as the preimage of a closed set. The last moment the car is not moving (i.e. the velocity is zero) is then well defined $$ \max v^{-1}(0) = \max\{t\in \mathbb{R}: v(t)=0\}, $$ assuming the car does not stop again later. Whereas the first moment the car is moving is ill defined as $v^{-1}(\mathbb{R}\setminus 0)$ is an open set as the preimage of an open set. It therefore has no minimum (first moment the car is moving). Its infimum is well defined and coincides with the last moment the car is moving.

Answer 11

Surely in order to define something as moving, there needs to be $2$ places at $2$ times? The whole concept of movement is a transfer from place $X$ to place $Y$.

If you measure an object at one time only, $t_1$, then you cannot know if it is moving or not. There is no speed, because $$\text{speed} = \frac{\text{distance}}{\text{time}},$$ and distance requires two places, $\mathrm{P_1}$ and $\mathrm{P_2}$

Or have I missed something?

Answer 12

For an object to be at rest one must be referring to some frame of reference.
A car may be stationary in frony of you while it is rotating around the earth's axis once per siderial day, orbiting the sun once per year, and orbiting the galactic centre. All these can be ignored by choosing a frame of reference where the car's location relative to you with time remains fixed.

If we ignore non-Newtonian mechanics, and it's reasonably safe to assume that your teacher was not intending to invoke relativity, then.

Velocity change = acceleration x time.

IF you treat the passage of time as linear and continuous then there is NO time period between being stationary and moving. If the teacher did not accept that answer then two alternatives come to mind. There will be others. ALL will be somewhat pedantic, but all will help expand your thinking processes :-).

Time is for practical purposes quantised.The shortest time period that 'exists' is Planck time = 5.39E-34 second. During this rather small period, if step pure force is applied unioformly to an "ideal car" then the state of motion of the car can not be determined until 1 Planck Time has elapsed.
Note that this is a simplistic and somewhat stupid answer but MAY be what he had in mind.

This brings us to another alternative.
I mentioned an "ideal car". They are hard to come by.
A non ideal car is compressible, and has distributed mass. Forces applied to it by gravity and propulsion system and reactive friction, stiction, aerodynamics once any part of it starts to "move" and more will mean, that when observed closely at very high frame rates, it "lurches" into motion. It will transition from being best described as stationary to being best described as moving. There is no certain absolute boundary if you insist on looking and measuring closely enough.
In practice such precise inspection is very very seldom of value, and we may say eg "when the wheels first start to move detectably" that it is in motion.

This also is only possibly hatw your teacher had in mind, but it has achieved the desired aim of getting you thinking.
And getting us thinking as well.
Your teacher would have been pleased :-) .

Answer 13

The question can be answered in a great many ways, and I feel sure the teacher was not looking for a "correct" answer but to see how you responded to it. I think he hoped you would challenge the question and that this would lead into a discussion of the importance of precise definitions. Most important to him might have been that cause and effect are never simultaneous, and that thought itself could be explored at many levels, as has been demonstrated in several answers.

Answer 14

Apparently the answer "the 2 moments coincide" was not a valid answer.

Lets consider the event ($x_0, t_0$) when the car was last stationary and ($x_1, t_1$) when the car was first moving. If $t_1 = t_0$ and $x1>x_0$ then we have $v = \delta x/\delta t = \infty$ which violates special relativity. If $x1=x_0$ then we have $v = \delta x/\delta t = 0/0$ which is indeterminate, but it also implies that event ($x_0, t_0$) is the same as event ($x_1, t_1$) and that they are in fact the same event and no motion has occurred. This is perhaps what your teacher was getting at when he said ""the 2 moments coincide" is not a valid answer. This paradoxical situation arises if we consider time and space to be infinitely divisible.

Calculus tells us we can assign an exact velocity value to an exact instant of time. That means we can ascribe a velocity of zero at instant zero and a non-zero value of velocity at a later instant. To say there is a zero interval between time zero and the next instant is to imply the velocity at time zero is both zero and non zero, which is illogical. If we assume a time $t_1$ which is the smallest time greater than $t_0$, we can always halve that value and find a smaller time. Let's assume at some small finite time, the time interval $\delta t_n$ is described by some fraction $\frac1{2^{n}}$, where n is an integer. As n goes to infinity, the interval goes to zero, but implicit in this assumption is that we can arrive at infinity by continuously doubling a number, which is of course nonsense.

In physical reality, one explanation is the Heisenberg uncertainty principle. This principle tells us the more certain we are about the momentum of a particle the less certain we can be about its position and vice versa. This means if we are absolutely certain a particle is stationary, we have a huge uncertainty in the particles position at time zero and if we are absolutely certain of a particle's location at time zero we have a huge uncertainty in its velocity. This make it impossible to say a particle is stationary at some exact instant, so the question in the title becomes almost impossible to define.

Almost all the paradoxes ascribed to Zeno are based on what would happen if we assume space and time are infinitely divisible. The Achilles and the tortoise paradox and the dichotomy paradox (and the question the OP) can be boiled down to whether or not the infinite sum $$\frac12 +\frac14+\frac18+\frac1{16}+ ...$$ is equal to 1. Almost universally, the modern interpretation is the infinite sum is exactly equal to one. It can be shown that for a finite series of terms the sum can be written as:

$$ S_n = 1 +\frac{1}{2^{n}}$$

For $n = \infty$ we get:

$$ S_{\infty} = 1 +\frac{1}{2^{\infty}} $$

and as n approaches infinity, the term $ \frac{1}{2^{\infty}} $ approaches 0 and so $S_n$ approaches 1.

Is this the same thing as saying the gap between one and the previous term is exactly zero and $S_n$ is exactly one?

The concept of surreal numbers would beg to differ. The Wikipedia article states (paraphrased) that infinitesimal numbers are smaller in absolute value than any positive real number. In other words there is a surreal number ($1/\infty$) between the smallest real number and zero and therefore the gap between the smallest real number and zero is not zero and this is basically the answer to 'What happens between the last moment the car is at rest and the first first moment the car moves?'. Also note that in the set of surreal numbers, the number $1/\infty$ is not equal to zero. In fact in this Numberphile video, Donald Knuth states there an infinite number of surreal numbers between any two real numbers, but what does he know?

Should we dismiss surreal numbers out of hand? After all, how can we discuss what is between the smallest real number and zero if we cannot define what the smallest real number is? Whatever we define it to be, we can always divide it half again and find a smaller real number. If I wanted to be awkward I could say, first tell me what the smallest positive real number is, before we discuss what is between the smallest positive real number and zero.

We can note that all of Zeno's paradoxes that involve motion or time, e.g. the Achilles and the tortoise paradox, the dichotomy paradox or even the arrow paradox, can be resolved if we assume there is an minimum indivisible discrete unit of time, possibly much smaller than the Planck interval. Unfortunately we are unable to probe down to that level and so there is no measurement we can perform to support or refute that idea.

Unfortunately I cannot provide a definitive answer to the OP and my personal belief is that there is no definitive answer, but I should stress that the majority accepted answer is that there is a definitive answer to the OP question (and Zeno's paradoxes) which is based on the premise that the infinite sum 1/2+1/4+1/8+1/16+... is exactly equal to 1 and that $1/\infty$ is exactly equal to zero, but to me, this implies that the gap between the smallest real number and zero is exactly zero. To me, the issue is not clear cut.

Answer 15

This question reveals a shortcoming of the naive zero-duration interactions between objects.

In the most naive model we have one block sitting still and another moving towards it. At some special moment $t_0$ they collide and there is a transfer of momentum. The velocities instantaneously change meaning the objects experience infinite acceleration for a zero-length duration.

Physics abhors infinities.

A more sophisticated model models the two blocks as squishy. They are like two springs. When the one box approaches the other, eventually it makes contact. The two springs compress a little bit over a finite amount of time, some momentum is transferred, then the blocks stop touching with the end result being a momentum transfer. The springs might ring a little bit even after the collision. This would be evidence of an inelastic collision.

But that model still has the problem where initially the springs are uncompressed and then instantaneously they feel a rapid change in their rate of compression. This is again an infinity.

A more sophisticated model considers the electromagnetic repulsion felt between the objects due to the objects that constitute them. In this case the force between the objects follows a Lennard-Jones potential so that even at large distances there is a super small but still non-zero force between the two objects. As they get closer the force gets larger until the distance between is on order of the interatomic spacing. Then the force rapidly, but smoothly, ramps up between the two objects. The lattices will compress some like in the previous model, and there will be a net transfer of momentum like in the original model.

It this model everything is smooth. There is no special moment $t_0$. There is a brief, but finite, period of time when the particles are experiencing a large repulsive force due to the repulsive part of the Lennard-Jones potential, but all of the physical variables evolve smoothly regardless of what special $t_0$ you might pick. There is no paradox. Just smooth Hamiltonian evolution.

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"Paradoxes" like these are usually resolved by remembering that nature abhors an infinity and figuring out how to make your model more sophisticated to get around the infinity.

Answer 16

As other answers have pointed out, this is an age-old form of query, but modern mathematics considers it to be a solved problem.

The most common answer I am aware of (and there are many different answers) is that there is no last moment the car is stationary. To make this argument, we start from the assumption that the motion of the car can be defined by function that accepts a real number, time, and outputs a boolean value, indicating whether it moves or not. Let $t_0$ be the first time where this function shows that the car moves. With this construction, we can define a Dedekind Cut, partitioning the real numbers into two subsets. One subset contains all times less than $t_0$ and the other contains the remaining real numbers. It can be shown that the first subset has no greatest element: there is no last time that a the car is stationary. This is provable with a proof by contradiction.

There are other answers. As one other answer here pointed out, one may disagree with the original assumption above. Instead, one may use another number system, such as the hyperreals, which leads to different answers,

Answer 17

I think the answer is called "wind-up", probably measured in milliseconds. When an automobile sits on a level surface, there exist various clearances between gears, universal joints and splines between the differential transmission and the driven axles of the vehicle. The crank shaft needs to make a partial revolution to take up the slack before the wheels start to turn. In addition, the torque applied to the drive shaft and other shafts in the power train adds another degree or so the rotation of the crank shaft to start the vehicle moving.

Answer 18

Are you sure the question was intended in such a philosophical manner? It can be deconstructed into ordinary high-school level physics:

Let's replace the car in our mind with any dumb block. The block "rests" (isn't moving in relation to our reference frame) either because there is no force applied to it at all, or because the force applied to it is less than the friction force. The applied force is not completely lost, it goes into internal deformations, even if we can't see it.

The applied force and the friction force are in static equilibrium. There is no notion of time, the force is either smaller than the friction, or it is larger.

Let's assume the force is continously increasing over time. Then there will be a momenent in time when the applied force overcomes the friction and the object would then start to be accelerated. Any infinitesimal amount of time after that leads to an increase in velocity v=a*t where the acceleration a comes from the applied Force a = (F_applied - F_roll)/m which is still reduced by the roll friction.

It felt to me that this was missing from the other answers. So again, time enters only because the force is changing. If it is not changing, the object/car either moves or doesn't move.

In the case of the car, it is the motor that is increasing the force applied to the wheels. There is no first moment of movement, any infinitesimal amount of time after the applied force has overcome the friction force results in a correspondingly small increase in velocity.

What happens between these 2 moments?
High-school level answer: The applied force has overcome the friction force.

Answer 19

The apparent contradiction arises because of the simplicity of the physical model when thinking of the car as a single object.

In reality, the car (or any object) is never at rest, when you consider all of the particles at atomic/sub-atomic level: they're all continuously moving in all directions randomly. For the entire system (the car) to move, a predominant direction has to appear in the randomness of atomic thermal vibration - and that happens gradually (i.e., from the very moment the clutch is engaged, or whatever system the vehicle is equipped with).

So gradually in fact, that in order to decide on the exact moment the movement starts, you need to decide on what level of analysis to apply to the system, and which exactly of the small perturbations initiates the change of velocity of "the car".

Answer 20

Most of the other answers are quite thoughtful and precise. No doubt, more educational than this one. Even so, perhaps a simpler interpretation could be useful.

Assuming the "calculus" and "differentiation" tags were not added by the author of the original post, this strikes me as, potentially, an introductory physics question. Consider a high school level physics course concerned with an introduction to Newtonian physics. The question may have been (albeit composed with some imprecision in language) attempting to extract a specific answer from the student. Perhaps it was merely designed to elicit a response such as: "A force was applied"

One thing taught in such a course is "Objects at rest tend to stay at rest". So how could the car's movement be explained if not for the application of a force?

All of this is speculation of course, based on my own experience in high school.

Answer 21

The above answers fail to remember that reality is quantized at teeny-tiny scales.

Even with ideal, platonic objects, if 2 of them touch and then later, they don't touch, THEN:

There exists a time such that the objects touched, then 1E-37 secs later, they do not touch anymore. That's called the plank time. Nothing can happen faster than that.

Zeno's paradox does not appply in physical reality because both space and time are quantized.

"The question is: what happens between these 2 moments?"

If the moments are 1 planck time apart, there IS no "between these 2 moments."