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I am learning about Instantaneous speed and velocity in Khan Academy. I understand the concept and calculation, but how do I know the different instantaneous speed at different points in real life?

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closed as unclear what you're asking by Bill N, ZeroTheHero, tom, sammy gerbil, Daniel Griscom Apr 4 '18 at 2:09

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    $\begingroup$ With speedometer $\endgroup$ – newera Apr 1 '18 at 1:45
  • $\begingroup$ Police measure the speed of cars with radar guns - but there are lots of different ways to do this - you should look up some yourself and then if you want ask a question which is more specific $\endgroup$ – tom Apr 3 '18 at 11:51
  • $\begingroup$ -1 Not clear what you are asking. Are you asking how you can measure instantaneous speed? And if so, are you asking about instruments or methods? $\endgroup$ – sammy gerbil Apr 3 '18 at 19:37
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The instantaneous speed can be measured using the doppler effect, which is how the radar guns works. It sends a pulse of light to a moving object, and when it gets reflected of the object and return, there will be a change in frequency given by $$\frac{\Delta f}{f}\frac{c}{2}=v$$

From this we can measure the instantaneous velocity of the object. But we cannot find it to absolute precision, due to the finite speed of light (Due to which this velocity we found would be that an infinitesimally small time ago) and also the uncertainty principle, due to which we cannot find with no error.

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Short answer: You can't.

Medium answer: You can't, it is a mathematical construct. However, we can measure average velocity, $\bar{v} = \frac{\Delta x}{\Delta t}$, that the difference between the measured average velocity and the instantaneous velocity doesn't matter to the answer of any questions you might care about.

Long answer: Any real measurement takes a finite amount of time and consists of discrete data points. As such it is technically impossible to measure the instantaneous velocity. However, we can do well enough. A realistic measurement of velocity may be done as follows. We can measure the position of an object, $x_i = x(t_i)$ periodically, sampling the position every $\Delta t$. We will then get a series of positions $x_i$ corresponding to times $t_i = t_0 + i \Delta t$. We can then calculate the average velocity during any time interval from $t_i$ to $t_{i+1}$ by

$$ v_i = \frac{x_{i+1} - x_i}{\Delta t} $$

This is not the instantaneous velocity but rather the average velocity over a time interval $\Delta t$. Now I need to justify to you why this can be "good enough" for a given experiment or calculation.

Imagine you are running up and down a hill. The hill is a 100 m long. Say you can run 5 m/s up the hill and 10 m/s down the hill. Then it will take you 20 s to run up the hill and 10 s to run down, 30 s total. Now let's imagine someone is measuring your velocity by measuring your position at various times separated by $\Delta t$ as described above.

Ok. First imagine they ONLY measure how long it takes you run the whole race. Their best estimate for your velocity during the whole race would be $v = \frac{200 \text{ m}}{30 \text{ s}} = 6.67 \frac{\text{m}}{\text{s}}$. This measurement is clearly missing some details because you were never actually running 6.67 m/s, You were running either 10 m/s or 5 m/s the whole time. The average is just an estimate.

Now imagine instead they measure your time at the top of the hill and at the bottom. Well this measurement would give them full details about your velocity because they would now know your average speed was 5 m/s up the hill and 10 m/s down the hill. So the point is if you have a shorter $\Delta t$ in between measurements you get finer resolution on the velocity and you can track changes in velocity that occur over the period of the measurement.

Let's now imagine the person measures your velocity every second. Well for the first 20 seconds they would always measure your velocity to be 5 m/s and for the last 10 seconds they would measure your velocity to be 10 m/s. However, they haven't gained any additional information over the previous case where they measured at the top and the bottom. That is, they are making measurements faster than your velocity is changing!!

This is the crux of my answer. You cannot actually measure instantaneous velocity, however, if you measure average velocity with a measurement every $\Delta t$, in such a way that $\Delta t$ is shorter than the time scale on which $v(t)$ varies, then you can get a "good enough" estimation of $v(t)$ for whatever application/experiment/problem you are considering. How do you determine what $\Delta t$ you should use? Well that depends on whatever thing you are measuring. For example, a runner probably doesn't change their pace very much over the course of 5 s so you could probably measure every 5 s and get a pretty good estimate of their instantaneous velocity. But now consider you were trying to measure the velocity of a pinball in a pinball machine. The pinball's velocity can undergo many major changes in the course of 5 s. This means you should measure its velocity much more often! Modern video uses a framerate of 25 frames per second. This means the $\Delta t$ between measurement would be 40 ms. You can imagine that this may still not even capture all of the motion of a pinball and you may instead want to use a high speed camera with $\Delta t$ of 1 ms or less!

One parting note. This concept of identifying appropriate scales for a given problem is something which is very imporant in physics in a much broader context than just measuring instantaneous velocity. Very often we approximate discrete series and continuous and vice versa. The license to make these approximations comes from identifying a smallest time scale, or smallest length scale or energy scale or whatever below which nothing in the problem is changing so it doesn't make a significant difference to the end result that we have made the approximation. This is a trick I highly recommend thinking about and becoming familiar with.

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You can never calculate an exact instantaneous velocity in real life. You can get an approximate value, but you can't get an exact one.

Two problems prevent this.

  1. Classically Lack of perfect knowledge

  2. The Uncertainty Principle

Classically I can never exactly measure any anything. I can make measurements relative to other things of know magnitude, but the limit of accuracy is always finite. I can never be exact.

The Uncertainty Principle is a principle from quantum mechanics. In simplistic terms it means I can't ever know both position and velocity exactly and if I know either exactly, the other value has an infinitely large margin of error.

In practice what you do with a macroscopic object is observe it and work out an velocity that fits a model of it's trajectory that you have.

Let's says I make a measurement of velocity $v_k\pm \delta v_k$ with an error range estimate of $\delta v_k$. How do I remove the error ?

Well I can't. I can only average it out. If I think the object is moving on a path $\vec{r(t)}$ whose forumla I know (but not the values of the parameters) I can try and fit my data to that formula and get approximate values for the parameters and use those to work out the instantaneous velocity using $\vec{v(t)}=\frac d {dt} \vec{r(t)}$. But there will always be some error because I did not have exact measurements of the velocities to fit the equations to. We can reduce the margin of error in our parameters by fitting more data to it, but it will never quite disappear.

The uncertainty principle (which doesn't matter much for large everyday scale objects) has a more extreme effect. It means that once I measure something like velocity, that measurement process itself will mean that I have changed the value to something else that is (more or less) random. The very act of measuring something changes it.

So at a quantum mechanical level, just trying to determine something's velocity will change the velocity to some unknown value. So the margin of error cannot be reduced by making repeated measurements, as all repeated measurements do is introduce more uncertainty, not less. And because uncertainty is measuring e.g. position and velocity (momentum) are linked, if I measure velocity, I've probably changed the path it's on to some new unknown path. So I can't even fit my motion to a fixed path using repeated measurements. Si quantum theory is fundamentally different in this sense.

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Here is another possible method with a couple of severe limitations, but probably acceptable for educational purposes: accelerometer.

If the initial speed of an object is zero (first limitation) and the accelerometer does not have any errors (second limitation), the accelerometer output will let you calculate a precise instantaneous speed of the object at any moment in time as:

enter image description here

I am not selling anything here, but if you can afford a 3-axis accelerometer, you can track instantaneous speed of an object in 3D space.

With real life accelerometers, the error accumulation will not allow you to measure the speed accurately beyond a short time window after the start.

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