# Force of a falling object.

Okay, I run an Emergency Services and Rescue Unit and was asked a very interesting question regarding our Roof Top Safety gear. A question I couldn't answer last night and I am desperate to find the correct answer.

The question posed was as follows:

How heavy would I weigh if I fell from a height of 3 meters and I weighed 95 kilograms (on Earth)?

If anyone could help out by putting up a forumla and the law that applies to it, it wold be greatly appreciated.

Cheers

Kris

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Your mass of 95kg would never change.

Your weight 95*g also wouldn't change over such a small difference.

Is your question more along the lines of "how much force would I impart on safety ropes (or gear) if I fell from that height and it caught me?"?

Hello Jared and yes, I was wondering what force is exacted onto the ropes and the like. Moreso I suppose my question is will the force that is applied to the rope exceed the SWL?

jtbell
Mentor
It all depends on how quickly the ropes or whatever bring you to a stop. The shorter the time interval, the larger the force required. Consider bungee-jumping off a bridge. The bungee cord brings you to a stop gradually as it stretches, and you don't feel much force. If you replace the bungee cord with an ordinary rope, you come to a stop very quickly at the bottom (assuming the rope doesn't break!), and you definitely feel it! (and probably break a few bones, too)

Well we could estimate a final speed you'd be travelling at after that height, but as per jtbell, it all comes down to how quickly the rope stops you.

We could assume the rope provides an instant stop (not a good idea because you enter the grounds of "short drop and a quick stop", potential hangman territory) and tell you the maxiimum force possible. If that's what you want? But again, if you want rope specifics there's a bit more detail to it.

Wow, so much to digest. lol I thought there was going to be a lot of stuff. Okay, so for arguments sake, lets say the rope is a static line with no stretch (sudden stop) what sort of force would be applied then? The second part to it would be what if that rope had the ability to stretch 18%?

I like Serena
Homework Helper
Wow, so much to digest. lol I thought there was going to be a lot of stuff. Okay, so for arguments sake, lets say the rope is a static line with no stretch (sudden stop) what sort of force would be applied then? The second part to it would be what if that rope had the ability to stretch 18%?
Your safety gear probably has specifications how much it can take.
I have to admit that I have never taken a look at such specifications, but perhaps you have those specifications?

Either way, I'm going to make an educated guess.

The safety gear would not so much take a force, but an impulse.
The impulse taken, is the difference between the momentum of your body before and after your fall.

When you fall a certain distance, your body has a certain speed.
In this case the momentum is given by the formula:

$$p = m v = m \sqrt(20 h)$$

where p is the momentum, m is the mass of your body, v is the speed of your body, and h is the height from which you are falling.

In your example we would have:

$$p = 95 kg \cdot \sqrt( 20 \frac m {s^2} \cdot 3 m ) = 736 \frac {kg \cdot m} {s}$$

Your safety gear would be designed to take this momentum and reduce it to zero.
To do this it exerts a force over a period of time.
The longer the time is to do this, the less force is needed to break your fall.
This is the heart of all safety gear, including seat belts, airbags and such.
They are designed to spread the time of impact over as large a period of time as possible.

The relation is given by the formula:

$$F_{average} = \frac p t$$

where F is the exerted force, p is again the momentum of your body, and t is the time during which your fall is broken.

If this time is for instance 1 second, this would mean the average force on your body is 736 N, which corresponds to the weight of a mass of 73.6 kg.
Since the real weight of your body is also carried, the total average force would be 950 N + 736 N = 1686 N, or the equivalent of 168.6 kg.

To get back to your question, a static line with no stretch (sudden stop) would give you a time t that is very, very short. This would imply that the force on your body becomes very great indeed, and might break your back.

I wouldn't be able to say how much force a line that would stretch 18% would give. For that I would need more information.
The best thing would be a specification of your safety gear that says for instance how much impulse or momentum it can take, or perhaps it gives some other set of parameters?

Thank you for your most informative reply., and yes to answer your question in relation to specifications on the rope, it does give safe working limit, but this only refers to the weight that can be applied to it gradually, not the weight if it is shock loaded. Thanks once again, but that pretty much clears up my question.

Cheers

Kris