Philosophy - #7 and 8

Braintwisters number 7 and 8 are what originally inspired me to write these posts. Both demonstrate a lack of understanding of math and physics.

Number 8 was the one that really made me do a headdesk. The "paradox" only occurs if one ignores velocity. At any given instance the arrow is in one specific place (as stated); however, it has a non-zero velocity!

A similar thing happens when you throw a ball in the air (sort of...I'm gonna take a derivative here...) When the ball reaches the peak of its flight, there's an instant where it hovers - it has zero velocity. But we know from experience that the ball will fall back to Earth. That's because the ball has a non-zero acceleration (in this case, it's the acceleration due to gravity: 9.8 m/s^2). If you look at a "snapshot" of the ball when it's at the peak, you might think that it will hover forever, but that's only because you're not looking at the full picture (so to speak). You've left out acceleration.

It's kind of the same thing here. The arrow is instantaneously in one place, but it keeps moving because of its velocity. If you don't include velocity you're not really looking at the whole system.

Likewise, #7 falls into a mathematical trap. It's essentially a related-rate problem (Calc 101 - Mom always said that calculus was useful). Achilles' velocity is greater than the tortoise's velocity. So eventually he will overtake the tortoise, even though the tortoise keeps moving. You can calculate the distance each has traveled (distance = rate x time) at any particular instant. At the time where distance_Achilles is greater than distance_tortoise, Achilles has passed the tortoise, even though distance_tortoise keeps increasing with time. Distance_Achilles increases with time, too - but at a faster rate.

Sorry if this was garbled; it's been a few years since I took Calc I and I'm not sure if I'm explaining it clearly. If you're interested, take a math class! Yay math!