Galileo’s Equal Time Ramps

April 24th, 2014

The time it takes for a ball to roll down a slope depends on the slope’s length and steepness. Galileo discovered an ingenious way to vary slope and steepness such that the descent time remains the same. Here, I go through the simple physics required to demonstrate that his method is correct.

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The Gamma Function – Factorials Unchained

April 8th, 2014

We all encounter factorials in high school math, including in the formulas for combinations, permutations, and the coefficients in the binomial theorem. We do not usually think of the factorial as being an example of a function. With an ordinary function, it is not restricted to having values only for positive integers, as is the case with factorials.

However, mathematicians have extended the concept of the factorial in such a way that we can find a factorial for any number. With this generalization, we can confidently say, for example, that -2.5)! = 2.36327…. It’s a bit like the way we extend the concept of exponents. A exponent that is a positive integer has an obvious meaning, but there is a need to generalize the exponent concept to include fractions and negative numbers.

In this post, I will describe this factorial function (which is actually called the Gamma function), plus a few uses for factorials in post high school math.

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Fabulous Formulas – the Rocket Equation

March 14th, 2014

Rockets were invented by the Chinese about 800 years ago, and were an early use of another Chinese invention: gunpowder. Considering that long history, it is surprising that it was not until 1903 that a physicist got around to finding the basic equation that lets us compute how a rocket will perform. That physicist was the Russian Konstantin Tsiolkovsky (1857 – 1935), the original rocket scientist.

The equation gives us the rocket’s velocity as a function of time, given four basic numbers that characterize the rocket. It is a pure application of Newton’s laws, and is fundamental to designing a rocket. In spite of its importance and interest, the high school physics textbooks that I have seen do not include it. This may be because the math behind it is not straightforward algebra. In fact, I have only seen it derived using calculus. In this post I will show how we can produce the equation without calculus, although there is a bit of hand waving involved in doing that.

After working out the equation, I will use it compute the final velocity of the Saturn V moon rocket.

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Standing Waves and String Vibration

January 22nd, 2014

As we all know, when a taunt string is plucked, it vibrates from side to side, producing a musical tone. In 1755, Daniel Bernoulli worked out the mathematics of vibrating strings. His goal was to determine the side-to-side displacement of the string at each point along its length as a function of time. This was a very new kind of problem. Each piece of the string must obey Newton’s laws of motion, but there are an infinite number of pieces, and they are interconnected.

Bernoulli’s solution predicted that there should be two simultaneous waves of displacement along the string, one travelling left, and the other to the right. The resulting string motion is the sum of these two waves.

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Fabulous Formulas: Kepler’s Third Law

January 15th, 2014

The animation shows the solar system, for Mercury through Mars. Obviously, the further a planet is from the Sun, the longer it takes to complete one orbit. However,the mathematical relationship between distance and orbit time is not obvious. If we make a list of orbit times and distance for the planets, one would be hard pressed to see how the two are connected, other than saying that orbit time goes up as distance increases.

Johannes Kepler was the first to try to find such a relationship. He struggled with the problem for more than 10 years, convinced that there must be formula that would connect the two. In 1619, he succeeded, and published what is now called Kepler’s Third Law.

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The Simple Physics of Rolling Objects

December 20th, 2013

If we roll a sphere,a cylinder, a hollow sphere, and a ring down a slope, they will arrive at the bottom in that order. Verifying this mathematically is an interesting and, I hope, instructive use of some basic physics, and I will do that in this post. Along the way, we will see that neither the size of an object, nor its mass, has an effect on the rolling speed. Only the type of shape matters.

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