# Man on ladder: jump off instantly vs last moment?

• vxr
In summary, there are two cases: free fall, which is a straightforward concept, and a ladder rotating and jumping off in the last moment. In this case, it is recommended to consider the forces acting on the ladder and the man, and to use energy calculations rather than equations for constant acceleration. It is also better to jump off at the last moment for the best outcome.
vxr
Homework Statement
A ladder of length ##l = 8.6 m## long and mass ##m = 60 kg## is placed in nearly vertical position against the wall of a building. You stand on a rung with your center of mass at the top of ladder. As you lean back slightly, the ladder begins to rotate about its base away from the wall. Is it better to quickly step off the ladder and drop to the ground or to hold onto the ladder and step off just before the top end hits the ground? In calculations take your mass.
Relevant Equations
##p = mv, E_{k} = \frac{mv^2}{2}##
So there are two cases:
a). free fall (straight forward for me)
b). ladder rotating and jumping off in last moment (I am interested in trying to understand this case)

I believe I should take into account momentum at the time the man hits the ground in both cases? The smaller, the better. Or should I take into account kinetic energy? Both use mass and velocity.

This is pretty much the b). case:

Red is the almost-vertical (assume vertical) ladder, and pink is its movement when it's rotating. Total distance the man standing on top of the rotating ladder travels is ##s = \frac{2\pi r}{4} = \frac{\pi r}{2}##

I need to find final velocity of such system. How do I find it? I think there is no horizontal acceleration, horizontal velocity is constant. On the other hand ##a = g## and vertical velocity is not constant.

Should I use this equation?

##v_{f}^2 = v_{i}^2 + 2as##

##v_{f}^2 = 2gs##

##v_{f} = \sqrt{2gs} = \sqrt{g\pi r}##

And when calculating the momentum I should take into account in this case sum of both ladder and a man's mass? Let ##M## be a mass of man.

##p_{f} = (M+m)v_{f}##

##p_{f} = (M+m)\sqrt{g \pi r}##

This seems too simple, I am sure I have made some mistakes, right? Perhaps forgot about some ##v_{x}## or ##v_{y}## components? And/or I shouldn't take into account sum of both masses, because after all the man jumps off of the ladder the very last moment.

Thanks for help.

Last edited:
Look up how rotating objects behave. Consider the ladder like a rod and find the moment of inertia. What forces are acting on the ladder and the man?

vxr said:
Should I use this equation? ##v_{f}^2 = v_{i}^2 + 2as##
No, that equation is only for constant acceleration. In this case you would need to replace "as" with an integral, and that is true whether you are considering the vertical component of motion or the tangential.
vxr said:
should take into account in this case sum of both ladder and a man's mass?
Only if you are going to allow the ladder to land on top of you.

@scottdave suggests considering the forces in rotational acceleration, but it is easier to work with energy. Consider what the KEs of the various components are at impact.

vxr
Without doing any proofs, it is better to jump off last moment, right?

vxr said:
Without doing any proofs, it is better to jump off last moment, right?
Yes.

vxr

## 1. Can jumping off a ladder at the last moment cause injury?

Yes, jumping off a ladder at the last moment can cause injury. This is because the sudden impact of your feet hitting the ground can put strain on your joints and muscles, especially if you are jumping from a high distance.

## 2. Is it safer to jump off a ladder instantly or at the last moment?

It is generally safer to jump off a ladder instantly rather than at the last moment. This is because jumping off a ladder at the last moment can increase the risk of injury due to the sudden impact on your body.

## 3. What are the potential risks of jumping off a ladder at the last moment?

The potential risks of jumping off a ladder at the last moment include sprains, strains, fractures, and other injuries to your feet, ankles, and legs. In some cases, these injuries may require medical attention.

## 4. Is there a proper technique for jumping off a ladder?

Yes, there is a proper technique for jumping off a ladder. This includes keeping your feet together, bending your knees, and rolling through the balls of your feet to absorb the impact of your landing.

## 5. What are some precautions to take when jumping off a ladder?

Some precautions to take when jumping off a ladder include making sure the ladder is secure and stable, using the proper ladder for the job, and maintaining three points of contact at all times while on the ladder. It is also important to avoid jumping from too high of a distance and to always jump with your feet together to maintain balance.

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