Our Last week's dog will reach the hare, how many jumps? Calling x the number of canine jumps and taking as unit of distance the hare jump, after those jumps the dog will have advanced 8x / 5, since 5 of his jumps equals 8 of the hare, and the hare will have advanced 4x / 3, since it gives 4 jumps while the dog gives 3. As the hare carries 60 of its advantages jumps, at the moment of the reach it will be 8x / 5 = 4x / 3 + 60, where x = 225. To reach the hare, the dog has to give 225 jumps, with which it will cover the 300 that the hare gives in the same time plus the 60 that it had of advantage.
The problem of the lifeguard and the bathers has provoked a wide controversy and interesting reflections (see comments from last week). To simplify, and since what is relevant is the relationship between speed on land and speed in water, consider that the lifeguard moves on land at 2 meters per second and in water at 1 m / s. If you go in a straight line to the bather, you will go rounding, rounding, 7 meters by land and others by water, which will take you about 10.5 seconds. If it runs (it is a saying) to the point of the edge closest to the bather, it will travel 11 meters and peak by land and 5 meters by water, with which the time employed will be practically the same. To improve this result, the lifeguard would have to behave like a ray of light. Why?
But the above is pure theory, of course: in real life, the lifeguard would go straight to the swimmer; thus, when diving, I would fully exploit the impulse of the race to the edge.
The problem of cars moving away on diverging roads has received almost no attention, so it remains pending.
The lifeguard and the light
When we talk about the speed of the light, normally we refer to its speed in a vacuum -which is usually represented by the letter c-, where it is maximum and approximately equal to 300,000 kilometers per second (exactly 299,792,458 m / s). Because it is one of the universal constants, we sometimes forget that the speed of light varies from one medium to another. For example, in water the light is somewhat slower than in the air, and that explains the well-known (but not always well understood) phenomenon of refraction (the well-known pencil that "bends" when immersed in a glass of water ).
Curiously, the way in which light is diverted when passing from air to water is expressed by a formula equivalent to that which determines the trajectory of the ideal lifeguard that attempts to reach the swimmer in the shortest possible time. Intelligent light or illuminated lifeguard? I invite my astute readers to explain in a simple way the phenomenon of refraction. There is no need to know more physics than the one we have just seen: that light travels in the water at a lower speed than in the air.
Carlo Frabetti He is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 scientific dissemination works for adults, children and young people, among themDamn physics,Damn mathematics or The big game. He was a screenwriter The Cristal ball.
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