The Crossed Ladders Problem

February 10th, 2016

Two ladders, one of length 100, and the other of length 80, lean against two walls, as shown in the diagram below. The point where they cross is 10 units above the ground.
What is the distance between the walls?
I have seen this problem, or variations of it, several times. Recently, I did some checking, and found that it appeared in Martin Gardner’s Mathematical Games column in Scientific American, June, 1970. The lengths in my diagram are the ones that Gardner used. In a subsequent column, he gave the answer, which is \(w = 79.1\). He said that the answer comes from solving a fourth degree equation, but he did not show a derivation of the equation.

I also found a Wikipedia article about the problem, which gives a derivation of what is probably Gardner’s method.

In this post, I will describe a solution of the problem that I came up with. It is not as elegant as Gardner’s method, but, to me in seems more straightforward.

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Newton’s Shells and the Earth Tunnel

October 28th, 2015

Isaac Newton is best known for his three laws of motion, and for the inverse square law of gravitation. Among the most important of his many other discoveries are two results that Sir Oliver Lodge called Newton’s Superb Theorems. They both have to do with the gravity force produced by a thin spherical shell.

Suppose the shell has a total mass of M and a radius of R. Its thickness is small enough that it can be neglected. If we were to place a test mass in the vicinity of such a shell, we could measure the net gravitational attraction from the shell. This gravity force would be the sum of the attractive force from each tiny piece of the shell. Each piece has its own distance from the test mass, and would add to the total attractive force according to the inverse square law of gravity. Newton wanted to find a formula for the total gravity force from the shell in terms of M and R.

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We Weigh Less on an Airplane

August 9th, 2015

Some time ago I realized that, for a couple of reasons, an object’s weight while aboard an airline flight would be slightly less than the object’s weight on the ground. Note that I am referring to weight, as measured by a scale, not the mass of the object — mass, of course, stays the same.
In this post, I will detail the two reasons for the weight reduction, and compute the amount of the effect at the typical speed and altitude of a commercial airliner.
I will then describe an experiment I did on an actual flight, where I was able to verify that the weight of a small object was indeed less than its ground weight.

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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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