Originally posted by TomCr
There is not really enough information here, so I will make some assumptions:
The ladder is standing at 90 degrees (it actually would be leaning against the wall at some undescribed angle).
The ground is 90 degrees from the top of the ladder.
The man moving down the ladder would not affect the speed of the ladder. This is, of course, a ba han this no matter how he does it.
This puzzle needs more information to do properly.
I did later say the ladder and the man massed 100 kg. Sorry, I made unstated assumptions about the setup, at first vertical with reference to the ground and the ground indeed 90 degrees from the vertical.
You state the man moves down the ladder at a uniform speed, don't you mean a uniform acceleration?
Also, it seems the man would be pushed away from the wall by the ladder on the way down, not much as I think he gets to the ground well before the ladder and the energy it takes to move the man away from the wall would be taken from the kinetic energy of the ladder's fall so that would complicate things a bit! It looks intuitively like the path the man would take, that is to say, the graph of his motion in Z and X would be a parabola.
As T=Sqr root (2S/A) for simple freefall time, 15 meter fall takes very close to 1.75 seconds. 30/9.8= 3.06, square root of that being 1.75 seconds, close enough anyway.
It looks to me like the X movement would be the half the ladders movement in 1.75 seconds and when the ladder is vertical, I think it safe to say for the initial velocity of 0.25 m/s while vertical, the assumption can be made to ignore gravity and just go with that initial push velocity.
So in 1.75 seconds, the top of the ladder would have moved 0.35 meter and as the man falls down he would be traversing sections of the ladder moving progressively less and less where at the bottom the movement in the X direction would be zero so it seems safe to say the ladder would have pushed him away from the wall half of 0.35 or 0.175 meter.
Does these seem accurate numbers? One assumption I made is to ignore friction between the man and the ladder on his downward traverse.
If those numbers are correct and he gets to the ground in less than 2 seconds, it seems for that amount of time he would be slowing down the fall of the ladder so it would have a skewed velocity curve on its downward fall, say 1.75 seconds of velocity A followed by strictly gravitational fall. I don't know how to calculate that one though.