I know what it means to say that my position is "X" at a particular moment in time. I can easily take a picture of my motion and observe my exact location at the instant the picture was taken. That is to say, my instantaneous position can be measured.

However, could I ever measure my velocity at an exact point in time? At best, it seems I can only estimate it based on the approximate derivative of my position or approximate integral of my acceleration over some measure of time.

Each of these methods, however, requires some elapse of time to perform the calculation, and even then it's only approximate. Neither allows me to know what my precise velocity is at an exact point in time, i.e. my instantaneous velocity is not being reported.

We obviously have devices like speedometers, tachometers, heat sensors, and dynos which ostensibly report quantities like velocity, heat flux, and power in real time. But really, how could they? Each of these quantities is itself the time derivative of another quantity and thus requires the passage of time to estimate (although velocity may be a bad example because we have things like pitot-static tubes).

Am I correct in surmising that rate-based instruments are not reporting in real-time? And if this is true, could we ever construct a device which did measure derivative quantities directly? What exactly would it measure?

  • $\begingroup$ Can I state the obvious? Every measurement must occur over a period of time. So we could never measure velocity at an exact point in time any more than we could measure position at an exact point in time. But you can get an instantaneous velocity by measuring it at two points in time! Apply the mean value theorem and say, "at some time between t=0.000 and t=0.001, the car was moving at exactly 20 meters per second". $\endgroup$ – user12029 Jul 15 '14 at 2:52
  • $\begingroup$ @NeuroFuzzy Distance is not measured over time, although it might take some time to do the measuring. $\endgroup$ – LDC3 Jul 15 '14 at 2:53
  • $\begingroup$ @NeuroFuzzy Try to calm down. Clearly if I have to measure two quantities at two points in time then I am not measuring the quantity between them, I am taking an average. I am asking what we can measure directly that is proportional to a rated quantity (like a pressure differential in a pitot-static tube). I am mostly interested in heat flux sensors and tachometers, as I do not know of any analogous quantity that can be directly measured (although I am learning through research). $\endgroup$ – Bryson S. Jul 15 '14 at 3:18
  • $\begingroup$ @BrysonS. But the two measurements can guarantee an exact velocity at some point in time within an interval. That's pretty good, and is contrary to the second sentence of your third paragraph. $\endgroup$ – user12029 Jul 15 '14 at 5:40

Years ago, the speedometer in a car moved the needle by spinning a magnet. The physical rotation of the driveshaft turned the cable inside the assembly. The spinning cable is attached to a magnet. The needle is mounted on a disk attached to a spring which provides rotational counter-force. The spinning magnet attempts to spin the disk, but the spring provides a force to stop the spinning. The faster the magnet spins, the more force is applied to the disk to spin it. See Wikipedia and the bottom of this page.

I believe this is a measurement of speed in real time since at any moment you can state your speed.

Also, the turning of the speedometer cable would turn the dial for the number of miles driven (which is why you could decrease the number of miles on the vehicle if you drove it backwards). This is a measurement of distance (the derivative of speed) in real time.

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  • $\begingroup$ Was the reported voltage directly produced by the RPM of the engine, or was it reporting the frequency of the current? $\endgroup$ – Bryson S. Jul 15 '14 at 3:20
  • $\begingroup$ @BrysonS. Neither. It was the physical rotation of the driveshaft that turned the cable inside the assembly. $\endgroup$ – LDC3 Jul 15 '14 at 3:22
  • $\begingroup$ But how does the cable motion translate into the position of the needle? One is a motion, the other a position... $\endgroup$ – Bryson S. Jul 15 '14 at 3:32
  • $\begingroup$ @BrysonS. The spinning cable is attached to a magnet. The needle is mounted on a disk attached to a spring which provides rotational counter-force. The spinning magnet attempts to spin the disk, but the spring provides a force to stop the spinning. The faster the magnet spins, the more force is applied to the disk to spin it. en.wikipedia.org/wiki/Speedometer#Eddy_current and the bottom of this page edn.com/electronics-news/4386069/… $\endgroup$ – LDC3 Jul 15 '14 at 3:53
  • $\begingroup$ This is really what I am after, the analog measurement of rates. The eddy-current setup is actually a perfect example of what I am talking about. Do you have any information about heat flux sensors? How do they measure heat flux (W/m^2) in analog? $\endgroup$ – Bryson S. Jul 15 '14 at 3:57

There are many devices that can measure rate of a quantity without using approximate derivative and integral. I will give some example here:

  • Pitot tube

    Pitot tube is a device used to measure velocity of a body with respect to flow. This device uses Bernoulli's equation. For computing the velocity using Pitot tube, the total pressure and static pressure must be known which these two quantity could be measeured instantaneously. Here you can find more detail on Pitot tube.

  • Orifice plate

    Orifice plate is used to measure mass or volumetric flow rate. This device also used Bernoulli's equation. For measuring mass flow rate (or volumetric rate) the pressure of the point before plate and the point after the plate must be known. Measuring these two pressure are very simple (e.g. by measuring the height of liquid column). More details can be found in many books and webpages (e.g. here).

  • Hot wire

    This device is used for measuring the velocity of a fluid flow. It used the laws of thermodynamics specially the first law and principles of electric circuits. Here you can find more details on this device.

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  • $\begingroup$ Yes I think I may have been swept up in the mindset of the digital age and forgotten about the historical, analog solutions to these types of problems. $\endgroup$ – Bryson S. Jul 15 '14 at 18:31

How about a radar gun? It works by measuring the Doppler shift of a radio signal incurred when it bounces off a moving object. The frequency shift of each photon encodes the instantaneous speed of the object when the photon scattered off of it. Difficult to get much more instantaneous than that (especially when you compare to a measurement of position, which would often be achieved by receiving and analysing some photons).

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  • $\begingroup$ Yeah I think we pretty much have our answer here, it's just the way I was thinking of the problem being immersed in the digital age and how many things are measured. We can measure rates in real time, but it requires some ingenuity... $\endgroup$ – Bryson S. Jul 15 '14 at 22:52
  • $\begingroup$ A Doppler shift requires the instrument to determine the frequency. The frequency cannot be determined with one point; at least 2 are required and to improve accuracy, more are needed. Since you cannot acquire the data instantly, this is not what the OP wants. $\endgroup$ – LDC3 Jul 16 '14 at 0:39
  • $\begingroup$ @LDC3 You do not need multiple points to measure a frequency (for light) because $E=h\nu$. With a sufficiently sensitive calorimeter, you can measure the energy of a single photon (in practice this works only for high energy photons, but there's no reason it couldn't work in principle). And so you also measure the frequency, with a single point. $\endgroup$ – Kyle Oman Jul 16 '14 at 3:49

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