Exploring the Physics of a Falling Ladder: Impact on a 200 Pound Man

[bpatterson9671]

This problem came up in discussion - If a 200 pound man was on the top of a 10 foot ladder, and the ladder fell in an arc to the ground, Would the force of impact on the man be less if he "road" the ladder down to the ground on the arc, or if he fell straight down 10 feet separate from the ladder? We are debating this at work and I am trying to come up with the physics behind this. Any help?

 

[OlderDan]

This problem came up in discussion - If a 200 pound man was on the top of a 10 foot ladder, and the ladder fell in an arc to the ground, Would the force of impact on the man be less if he "road" the ladder down to the ground on the arc, or if he fell straight down 10 feet separate from the ladder? We are debating this at work and I am trying to come up with the physics behind this. Any help?

If you ignore the mass of the ladder, you can treat this as a simple conservation problem. What is conserved? What is the velocity of the poor guy (magnitude and direction) when he hits the ground? Which component(s) of his velocity will result in doing him harm?

 

[bpatterson9671]

I would ignore the mass of the ladder (lets assume it is a really light aluminum one), and I have the equation F = mv , however, I am having a difficult time with the arc velocity calculation. I can calculate the straight line fall due to gravity, but I am not sure of the other.

 

[OlderDan]

I would ignore the mass of the ladder (lets assume it is a really light aluminum one), and I have the equation F = mv , however, I am having a difficult time with the arc velocity calculation. I can calculate the straight line fall due to gravity, but I am not sure of the other.

The force the ladder exerts on the man is perpendicular to his motion. That makes this problem like masses sliding down surfaces, plane or otherwise, roller coasters, pendulums, etc. Mechanical energy is conserved. You can find the velocity at any height using energy conservation. You can find the direction of the velocity using the motion of the ladder.

If you wanted to be a bit more precise, you would recognize that even if the bottom of the ladder were against a wall, at some angle the ladder would lose contact with the wall and start moving horizontally. The man's path would change from circular to parabolic. Assume first that the bottom of the ladder is constrained to not move, only pivot. Add the possibility of it moving later if you want.