# Is instantaneous velocity an abstraction?

In introductory analysis, the discussion the derivative emphasizes that while average rates of change are measurable, instantaneous rates of change are a "limiting abstraction". While this makes perfect sense from the formal view of analysis, I wonder how it maps to the view of physisists.

Are instantaneous rates of change, such as velocity, only ever inferred from observable average rates of change (measured or imagined over arbitrarily small intervals), or can they be directly observed?

• Think of the Doppler effect. There, it is the speed that determines the shift in frequency of the wave phenomenon. So, as you measure the frequency, you can acquire knowledge of the instantaneous speed. – Raskolnikov Nov 10 '11 at 15:05

## 2 Answers

Instantaneous velocity can never be measured since there is no way in the real world to do anything instantaneously. All measurements take some amount of time to peform.

For example the comment to the question mentioned using the Doppler effect to measure instantaneous velocity. That is not possible since to measure the frequency of a wave you have to observe it for something on the order of a cycle of the wave, so the Doppler effect will only ever measure the average speed of the object over some small time interval.

Quantum mechanically speaking...

Objects only have instantaneous velocity insofar as they don't have instantaneous position and mass. Velocity is merely the ratio of momentum to mass; the uncertainty can be calculated.

We have $p = mv$; from the rules of propagation of uncertainty we get

$$\sigma_p = mv \sqrt{\left(\frac{\sigma_m}{m}\right)^2 + \left(\frac{\sigma_v}{v}\right)^2 + 2\frac{\mbox{COV}(m,v)}{mv}}$$

where $\mbox{COV}(m,v)$ is the covariance between the mass and velocity. I am confident that this term can be worked out explicitly, but it has been a couple years since I took quantum mechanics and I don't have the time to do it right now. It should also be possible to write the mass in terms of the velocity depending on what you know about the system and observer at least. You can then substitute this into the uncertainty equation

$$\sigma_x \sigma_p \geq \frac{\hbar}{2}$$

This will set a limit on the uncertainty of the velocity in terms of the uncertainties of mass and position. Clearly, however, the uncertainty cannot go to zero and the ultimate resolution of your question then comes down to your definition of "instantaneous."