Problems with Slipping Ladders

A ladder leans against a wall. It begins to slide, the top end moving down the wall and the bottom end across the floor away from the wall.

Such a scenario is the basis for a variety of mathematics problems. For example, if the ladder's bottom end moves away from the wall at a constant speed, what is the velocity of the top of the ladder at any given instant? Curiously, the mathematical model indicates that the velocity of the top approaches infinity when the ladder hits the floor.


How can that be? Even with a smooth wall and a frictionless floor, conservation of energy alone dictates that the velocity can't approach infinity. The answer lies not in the mathematics, which is correct, but in the physics of the ladder's motion.

It turns out that the ladder loses contact with the wall when the top has slid one-third of the way down. The original equations describing the situation no longer apply. The ladder then continues to move away from the wall, in a motion determined only by gravity and the reaction force of the floor. A different set of equations describes this part of the motion.

Stelios Kapranidis and Reginald Koo of the University of South Carolina Aiken present this scenario in their article "Variations of the Sliding Ladder Problem," published in the November College Mathematics Journal.

"Calculus teachers are familiar with the sliding ladder problem from the study of related rates," Kapranidis and Koo write. "However, it is not well known that some versions of this problem are physically impossible."

What happens when the ladder is somehow linked to the wall and can't leave it as it slides down? In this case, the velocity of the bottom increases to a maximum (when the ladder is one-third of the way down the wall), then decreases to a zero by the time the ladder hits the floor. The velocity of the top increases from zero to some finite value.

What conditions do you need for the top of the ladder to maintain contact with the wall and the bottom to move at a constant speed. Kapranidis and Koo note that such a motion can be realized by introducing extra forces to pull down on the top and push back at the bottom.

"This situation does yield arbitrarily large speeds," the authors say. "Achieving this motion requires the application of constraint forces that approach infinity." In other words, this motion can't happen.

"It is only when we consider the forces that cause the motion that we can resolve the paradox of speeds approaching infinity," Kapranidis and Koo conclude. "They also indicate caution in regarding the sliding ladder as an example of a real world application of mathematics."

References:

Freeman, M., and P. Palffy-Muhoray. 1985. On mathematical and physical ladders. American Journal of Physics 53:276-277.

Kapranidis, S., and R. Koo. 2008. Variations of the sliding ladder problem. College Mathematics Journal 39(November):374-379.

Scholten, P., and A. Simoson. 1996. The falling ladder paradox. College Mathematics Journal 27(January):49-54.