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When speaking of the velocity of a particle, we must specify a reference point. The same holds when we talk about the speed of a particle (is this correct?). If so, a speedometer measures instaneous speed relative to what reference point?

Also, suppose we have car particle moving a circle at constant angular velocity. Note that the both the speed recorded by the speedometer AND the speed of the car relative to the center of the circle is constant.

Now instead suppose we have a car moving along a parabolic path (Ex: y=x^2) such that the speedometer records a constant speed. In this case the speed of the car relative to the origin is not constant.

It seems that the speedometer does not seem to measure speed relative to any fixed reference point, instead it measures derivative of the arc length traversed on the path with respect to time. Is this conclusion correct? If so does this means that speeds can be absolute, meaning they are the same not depending on the reference point?

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A car's speedometer typically measures the frequency of rotation for one of the wheels; in a two-wheel drive vehicle it's usually one of the non-drive wheels. The calibration to highway speed is based on the recommended tire size for that wheel. In many vehicles the speedometer counts only integer rotations of the wheel, and is therefore unreliable for speeds below about 10 mph.

It's possible to miscalibrate the speedometer in several ways: for example, by putting incorrectly-sized tires on a car. A car driven at a particular center-of-mass speed round a right-hand turn will report a different speed than around a left-hand turn, since the inside tire track of a turn is shorter than the outside tire track. And a car which is sliding or in another situation where the non-drive wheels are not turning may report that it is not moving at all; this is an important plot point in the film Ferris Bueller's Day Off.

Your statement that the speedometer measures the time derivative of the arc length traversed by the car is, in the limit of non-slipping tires, correct. It doesn't mean that speeds are absolute, because the speed is (hopefully!) relative to the road.

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    $\begingroup$ In fact the speedo drive is normally taken from the gearbox output shaft. This means that the differential gear will average out the effect on wheel rotation speed of driving around a bend. $\endgroup$
    – peterG
    Commented Mar 2, 2015 at 14:28
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The speedometer of a car measures the rate of rotation of wheel in a definite interval of time

It also measures the average speed/instantaneous speed relative tn reference point.

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