Problem in understanding instantaneous velocity

[jbriggs444]

Sure you can, and speedometers do exactly this. One kind of speedometer on a motorcycle I own has a gear driven sensor on the front wheel. The sensor has a worm gear that turns when the wheel turns. The worm gear drives a cable, the other end of which causes a magnet to rotate that in turn causes a needle to sweep to a certain position that corresponds to the speed of the motorcycle (speed = magnitude of velocity). Another motorcycle I have has a cable that comes from the transmission that drives the speedometer in a similar way. These measured values are the instantaneous speeds.

The needle has mass and is damped. Undamped needles bounce. A torque that acts on such a needle will not result in an instantaneous change in the needle's position. What the speedometer shows is some sort of approximate weighted average of past velocities.

 

[Mike_bb]

The needle has mass and is damped. Undamped needles bounce. A torque that acts on such a needle will not result in an instantaneous change in the needle's position. What the speedometer shows is some sort of approximate weighted average of past velocities.

I read more about speedometers and I agree with you. Only analogue speedometer with the arrow can show exact value of instantaneous speed because such speedometer is based on induction principle.

 

[Mark44]

The needle has mass and is damped.

Only analogue speedometer with the arrow can show exact value of instantaneous speed because such speedometer is based on induction principle.

The speedometers on my old bikes are strictly analogue, and as far as I know, aren't damped.

 

[robphy]

Here's how I visually motivate the interpretation of the velocity (as the slope of the tangent line) to my students.
Assuming a nice x-vs-t graph,
at the instant T of interest, centered at (T,x(T)),
I zoom in enough so that "my graph looks like a straight-line" in the viewport.
Since the motion is practically a steady-velocity motion for a sufficiently-short time-interval,
the [instantaneous] velocity is practically equal to the slope of that [approximate] straight-line in the viewport.

In the Desmos visualization below,
I have already activated the tangent-line (which you can disable by clicking on the filled circle for its folder).
To animate the zoom, click on the play-button ⏵ of the z-slider. (You can drag the z-slider to manual control the zoom.)

www.desmos.com/calculator/ghfds0lbht