Even though it was now middle of winter, Jack hauled out his ladder and placed his ladder against the side of the house and began to climb. He had some bad luck when he reached the halfway point. The ground was a frozen sheet of ice and the base of his ladder slipped out and the top slid down the side of the house. Jack, clinging to the center rung, wound up moving from Point A on the side of his house to Point B on the ground.
Describe the path Jack traveled.
The path traveled would be a smooth arc. Assuming he climbed up directly against the wall  though this does not matter since the question does not ask this  he travels from point A on the wall to point B on the ground.
The ladder, being a straight line, can be viewed to fall in an infinite number of very short time lengths. In each of these lengths, the ladder itself forms a tangent line to the actual path Jack takes from point A to point B. The ladder itself is of fixed length, and the bottom of it always touches the ground. This can be viewed an the X axis. With each successive time instant, the distance from Jack to the ground decreases, and hence the angle increases. All of this points to the arctangent path.

Posted by Eric
on 20040727 11:29:52 