Let the guard be a distance "b" from the edge of the beach, a horizontal distance "h" from where the drowning guy is, and let the drowning guy be a distance "d" from the shoreline.
Finally, let the lifeguard run to a point "x" on the shore before entering the water. See picture:
Then the time it takes him to get out there is
We want to find "x" that minimizes this time, so take a derivative and set it equal to zero.
Now take another look at the geometry of the problem. The expression above can be rewritten as
which is our answer: Snell's Law. Notice that all the actual distances fall out - only two of their proportions matter.
Square Root Calculator
The Exploratorium in San Francisco has a device they call a "square root calculator". It's just a ramp, a ball, a jump, and a landing zone, like this:
How is this a square root calculator? How does it work? Can you suggest another simple physical system that would operate as an multiplication calculator, logarithm calculator, trigonometric calculator, etc?
5 comments:
I'm assuming that once the ball hits the end of the ramp, the potential energy stored from its height is transferred into kinetic energy propelling it horizontally. The earth's gravity accelerates the ball as it flies off the end of the ramp down to the ground equally in all cases, so really the main variable is the distance it travels horizontally.
mgh=1/2mv^2, or something like that. You calibrate the spacings correctly, and you'll get that sqrt(h)=v
Does that solve it?
Why did my solution to this problem disappear?
I posted on this yesterday.
And now it retrieved one comment.
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