>Model cars use RFID tags for lap timing and seem quite reliable.

I suspect that this is how the formula one and similar race cars do it as
well. However in these cases I suspect that it is a permanently transmitting
beacon seeing power is available.

For the OP, is it not possible to have a slot cut in the track to drop a
wire receiver aerial into, or is it a dirt/cinder track? If a hard surface
then potentially any other uses of the track would probably also welcome
having the receiver loop permanently embedded, even if each club had to own
its own timing gear (maybe an opportunity to hire the gear you own).

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