### One of a continuing series

This is not my original. I pulled from a book I’ve had for about 60 years. See the diagram above, also from the book. Here’s the Quiz Question:

SUPPOSE a train travels in a finite time from station A to station B along a straight section of track. The journey need not be of uniform speed or acceleration. The train may act in any manner, speeding up, slowing down, coming to a halt, or even backing up for a while, before reaching B. But the exact motion of the train is supposed to be known in advance; that is, the function s = f{t) is given, where s is the distance of the train from station A, and t is the time, measured from the instant of departure. On the floor of one of the cars a rod is pivoted so that it may move without friction either forward or backward until it touches the floor. (If it does touch the floor, we assume that it remains on the floor henceforth; this will be the case if the rod does not bounce.) We ask if it is possible to place the rod in such a position that if it is released at the instant when the train starts and allowed to move solely under the influence of gravity and the motion of the train, it will not fall to the floor during the entire journey from A to B.

Also answer why. Provide your answer as a comment below.

## Update and hint

This is a famous math problem. It’s called the Lever of Mahomet, and I took it from a volume of The World of Mathematics. You have to come up with the answer without looking it up. Here’s the hint. Think continuous functions and what that implies.

## Update and answer

As promised, I’m posting the answer today. There have been some comment, on and off line. It’s time to call the play.

This is one of those rare times when your intuition was spot on. You should listen to the inner voice more often. It goes like this:

- Look at the diagram. If you place the bar too far to the left, it will fall to the floor on that side no matter what movements are made by the car.
- Similarly for the right.
- Since the movement of the car is a continuous function of time, there must be a point between these two extremes where the bar will still be off the floor when the car reaches B. After that, the bar can fall to the floor.

You can find a discussion of the problem on page 2389 of *The World of Mathematics*. Richard Courant and Herbert Robbins were the original authors. They provide a comprehensive analysis.

Yes. The problem is symmetric. Accelerate to the halfway point and then decelerate to the end. With a little “burst” at the halfway point to “reverse” the position of the rod.

Mike has posted a comment, but it does not address the central problem. Mike’s comment proposes a motion for the car. But the motion of the car, while known in advance, is not of his choosing. The question is whether for a given function of motion, is there always a starting position of the rod so that the rod does not touch the floor between positions A and B.

Yes, there are. Let’s say the train is moving at a constant speed between A and B, the roof could remain at precisely 90 Degrees (vertical). If it is decelerating, it would should be leaving away from the direction of motion. If it is accelerating all through them it should be leaving into the direction of motion.

Depending on the function (s) we can calculate the starting position of the rod (the angle).

That said, given all or some actual values of the parameters it would be harder, if not impossible, to find a solution for a function. Provided we can manipulate some of them we should find the actual position of the rod that won’t hit the floor during the journey.

Prasad,

Read the problem again. You are not asked to derive a solution. What is required is to say whether or not a solution exists.

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