Finish Speed versus Arrival Time

OKAY, BUT… HOW? · A closer look

Question: Can two journeys end at the same speed but take different amounts of time?

You’ll be able to: Distinguish speed at the finish from elapsed time over the whole route.

Finish speed asks how fast the marble is moving at one place. Arrival time asks how long the whole journey took, from release to that place. A finish-speed comparison cannot tell you every part of the journey.

For the same marble let go without a push, rolling without skidding and with no energy losses, equal drops give equal ideal finish speeds. Different route shapes can still produce different journey times. Time depends on how the marble moves all along the route.

Harvard describes a ball race in which a curved route beats a straight incline. That is a specific demonstration, not a rule that every curve wins. A homemade track needs timing rather than a guess based on its length.

Diagram in words: Two stacked route panels have equal relative endpoint heights but different downhill profiles. A separate timing strip marks release to finish, while separate text identifies speed at the finish. No times or winning route are claimed.

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Try the idea

You know two ideal runs have equal finish speeds. Can you announce a tie in arrival time?

Answer: No. Equal endpoint speed does not tell you the speeds throughout either route. You need the journey times to decide whether they tie.

A little further, if you want

Optional: Illinois derives the time distinction for a particle sliding without friction. That is not the spinning-marble speed model. It illustrates why a whole-path question needs more information than endpoint heights.

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Sources

• Harvard: Brachistochrone demonstration — A described curved-versus-straight ball race.
• Illinois: Brachistochrone teaching notes — Whole-route time in a frictionless sliding-particle model.
• OpenStax: Rolling Motion — Ideal rolling energy and real transfers.

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