When should you jump off a falling ladder?

Question

If you stand on the top of a falling ladder you will hit the ground at a higher speed (and therefore presumedly sustain more injury) if you hold on to the ladder than if you jump off it. This was solved here.

Where is the "break even" height on the ladder, from where you will hit the ground with the same speed if you jump off it or if you follow it down? This question just makes an assumption that you would hit the ground more softly if you stay on to the ladder (compared to jumping off it), if you are located at the lower part of it.

I don't think the midpoint is the break even point. I quickly calculated and I think you should stay on the ladder if you are on its midpoint. (The following is just a quick computation, there could be errors in it.)

$$ v_{midpoint}^2 = \frac{1}{2}gl \frac{m + m_L}{\frac{1}{4}m + \frac{1}{3}m_L} $$

$m$ and $m_L$ are the respective masses of man and ladder, $l$ is the length of the ladder.

Answer 1

You should stand at 2/3 of the height of the ladder.

If you land with the same kinetic energy as without a ladder, then the ladder should land with the same kinetic energy as without you. Equating the kinetic energy of the ladder with its potential energy at the beginning:

$$\frac{1}{2} mgL = \frac{1}{2} I_L \omega^2 = \frac{1}{2} \left(\frac{1}{3} mL^2\right) \omega^2$$ gives: $$\omega = \sqrt{\frac{3g}{L}}$$

where $L$ is the length, $m$ the mass, $I_L$ the moment of inertia and $\omega$ is the angular velocity of the ladder.

For you the same equation holds, but now $\omega$ is known:

$$MgH = \frac{1}{2} I_M \omega^2 = \frac{1}{2} (MH^2) \left(\frac{3g}{L}\right)$$ with $M$ your mass, $I_M$ your moment of inertia and $H$ your height. Solving for $H$ gives:

$$H=\frac{2}{3}L$$

or of course $H=0\;.$

Answer 2

We can do this with a minor modification to the calculation described in the earlier question. As before we'll take the ladder length to be $\ell$, but now we'll take your height to be $\alpha\ell$, where $\alpha$ ranges from zero to one. Our reference point is if you let go, in which case your speed when you hit the ground will be:

$$ v^2 = 2g\alpha\ell \tag{1} $$

Now suppose you hold onto the ladder. As before we calculate the total potential energy change of both you and the ladder, which is:

$$ V = mg\alpha\ell + \frac{1}{2}m_Lg\ell \tag{2} $$

And this must be equal to the increase in angular kinetic energy $\tfrac{1}{2}I\omega^2$. The combined moment of inertia of you and ladder is:

$$ I = m(\alpha\ell)^2 + \frac{1}{3}m_L\ell^2 $$

And setting the kinetic energy equal to the potential energy gives:

$$ mg\alpha\ell + \frac{1}{2}m_Lg\ell = \tfrac{1}{2}\left(m\alpha^2 + \frac{m_L}{3}\right)\ell^2\omega^2 $$

And since $v=r\omega$ your velocity is $v=\alpha\ell\omega$ giving:

$$ mg\alpha\ell + \frac{1}{2}m_Lg\ell = \tfrac{1}{2}\left(m\alpha^2 + \frac{m_L}{3}\right)\ell^2\frac{v^2}{\alpha^2\ell^2} $$

Which rearranges to:

$$ v^2 = g\ell\alpha^2 \frac{2m\alpha + m_L}{m\alpha^2 + \frac{m_L}{3}} $$

And finally substitute for $v$ from equation (1) to get:

$$ 2g\alpha\ell = g\ell\alpha^2 \frac{2m\alpha + m_L}{m\alpha^2 + \frac{m_L}{3}} $$

And this rearranges to:

$$ \alpha = \frac{2}{3} $$

So if you are more than $\tfrac{2}{3}$ of the way up the ladder you should let go, while if you are lower than $\tfrac{2}{3}$ of the way up the ladder you should hang on.

Answer 3

If one has the presence of mind, I vote for sliding down the ladder at any cost - gloves and shoes acting as a control brake. Shortening the diameter (height of your feet from ground) will surely reduce any impact. The risk is loss of control in the vertical decent. Gloves might be a recommended safety feature of any ladder climbing to keep confidence while executing a rapid slide. I would think you could get 6 feet lower before getting into the lateral arc. I admit that my own experience is limited to sliding off a wet cedar shingle roof and kicking the ladder away as I slid into it. It was only 10 feet onto damp lawn so I was saved by that and walked away. Yeah, I've had the training but I landed on my butt. No roll was possible.

Answer 4

I think that staying with the ladder by locking yourself to the ladder with a very strong grip is recommended because not all ladder falls complete to the ground. The fall could be stopped by another object such as a vehicle, a tree or a wall.

Answer 5

You people are crazy with your useless calculations.

Doesnt take science to realise that if you hold on the ladder you will have a higher chances of falling on your back or ass and hit your head against the ground now I wouldnt know peoples preferences but yesterday I was at the top of the ladder which is about 3 meters I was trying to screw a piece of mdf sheet when I lost balance and gravity was pulling me backwards so I simply jumped off It and landed on my legs, now my left leg heel is killing me when I step on it but I rather have it this way than ended up in hospital with a concussion or broken back. Ps; the floor was concrete.

So yeah no matter your calculations it's better to jump and fall on your legs than hit any other part of the body, trust me.