How fat DOES a man have to be to stop a runaway trolley?
October 4, 2012 2:21 PM   Subscribe

How fat DOES a man have to be to stop a runaway trolley? Check my math/physics please.

I wanted to make an animation exploring a trolley problem from a physics standpoint rather than a moral philosophy standpoint, inspired by some of the comments here particularly ROU_Xenophobe's.

Now my brain is fried trying to figure it out.

I figured I'd keep it simple (but not simple enough) by focusing on a scenario where the trolley pushes the man in front of it, and the increased friction stops the trolley.

Assuming a 15000lb trolley moving at 15mph a required stopping distance of 300 yards and an assumed kinetic coefficient of friction of 0.68 I'm coming up with a 184 pound man, which is a lot smaller than I intuitively thought, so I figured I should double-check I'm not making a dumb mistake.

Assuming I convert them all into kg metres/second and metres

(speed^2*mass)/(distance*2) = Force in Newtons required to stop it

And then convert the newtons to pounds, and apply the coefficient of friction to get what he has to weigh to add that force from his friction.

So I combine all that with the metric conversions, and I get something like this:

Weight in lbs = ( 15mph * 0.4470 mps/mph )^2 *15000lb * 0.4535 kg/lb * 0.2248lb/N / ( 300yd * 0.9144m/yd *2 *0.68 )

Which comes out to only a 184lb man.

Check that I have correctly used formulas for stopping and force, please, though I will also entertain discussion of what other factors I'm ignoring if they would change the result drastically.

For the coefficient of friction I used an example I found for "tires on a dirt road" because I didn't feel like dragging someone along the ground. And my other assumptions were to give the possibility of stopping it as much credence as I could.
posted by RobotHero to Science & Nature (10 answers total) 5 users marked this as a favorite
The value for tires on a dirt road would be static friction - the force required to start moving a stationary object. For an already moving object, the coefficient of friction is going to be lower.

Also, I expect the durability of the object in question is probably the real question. A person is going to be losing mass rapidly due to being turned into a bloody streak.

Also, 15 miles an hour is still slow enough where most people can probably jump from the trolley and survive without critical injury, so throwing a man in front of it may not be the only option anyway. Maybe the doors could be locked or something.

At any rate, it's probably easier to come up with a more plausible scenario for moral philosophizing than try to come up with a situation where a popular (but silly) one might happen.
posted by Zalzidrax at 3:14 PM on October 4, 2012

Response by poster: Where's the fun in coming up with a plausible scenario or better strategy?
posted by RobotHero at 4:50 PM on October 4, 2012 [1 favorite]

I don't really remember the physics because I didn't do my homework that day but I don't quite get how you're trying to calculate this. If you had an inflexible, indestructible object like Captain America's shield and it was wedged between two railroad ties, it would just stop the train even though it weighs almost nothing. Or if you're conceiving of the fat guy as functioning like the brake pads on a car, their mass still doesn't matter at all.

I think you might want to be looking at elastic and inelastic collisions. But even if you take the example of a nearly-elastic collision where no energy is lost to deformation like a game of pool, you need a pool ball of the same mass as the cue ball to completely stop it. Imagine if you shot a regular ivory or plastic cue ball at a ball the same size but made of styrofoam or balsa wood; it wouldn't stop.
posted by XMLicious at 6:30 PM on October 4, 2012

Response by poster: If it was purely elastic and the fat man was the same weight as the trolley, the trolley would come to a dead stop and the fat man would start moving at 15mph. And then your guys would just get run over by a 15000 pound fat man instead of a trolley, unless some other force, like friction, stopped him first.

If it were an inelastic collision, you would then have 30000lbs of combined trolley and fat man traveling at 7.5mph that you would still need to stop.

So that's why I was focused on the friction.

I had pictured the man being pushed in front of the trolley and dragging on the ground. That dragging would cause friction and apply a constant force to stop the trolley. The amount of that force will be his own weight multiplied by the coefficient of friction, whatever it is.

If I want to stop it in distance D and it's traveling at speed S and it weighs M, then what is the force, F, I need to stop it?

The time it takes to stop will be the distance divided by the average speed. The average speed will be half the starting speed, since it goes from 15 to 0.
T = D / ( S / 2 )
To get from speed S m/s to 0 m/s in time T s, I need acceleration A m/s^2.
A = S / T
If I substitute the first formula for T, I get
A = S / (D/(S/2))
And if I simplify that, I get
A = S*S/(2*D)

And of course, Acceleration is Force divided by Mass,
And I solve for F
A*M = F
And substitute the above for A
S*S*M/(D*2) = F

(speed^2*mass)/(distance*2) = Force

I had done all this and got a much higher number, and done most of my animation when I realized I had made a dumb mistake in converting Newtons back to Pounds. And then when I converted correctly, my man wasn't even fat anymore!

So I thought I should put it up here in case I've done anything else wrong.

The idea of him losing mass from his flesh being ground off is an interesting possibility, but unless there's a simple way to estimate how quickly he'd lose mass as he's dragged, I'm going to ignore that element of the model.
posted by RobotHero at 8:14 PM on October 4, 2012

You are giving yourself 300 yards to stop a relatively light trolley. That is quite a long way. I don't know if you've ever been sailing, but you can stand on a dock and grab a 10,000lb sailboat moving at 1mph and stop it by pulling in the opposite direction in about two steps worth of distance.
posted by tylerkaraszewski at 8:39 PM on October 4, 2012

Response by poster: Hmm. I'll have to re-write the ending to reflect the new numbers.

The Toronto trolleys are still 81000 pounds, and if you use one of those moving at 25mph, then he needs to be 2765 pounds, which gets to be a fat man, again. (Assuming the rest of my math still holds?)

The 15000 pound cable cars are acknowledged as the exception, rather than the rule.
posted by RobotHero at 10:36 PM on October 4, 2012

Alternatively, what's the spring constant of flesh? Solving the problem using Hooke's law would make for a better animation, I'd think.
posted by GenericUser at 1:36 AM on October 5, 2012

I'm pretty sure that for any non 'super' type humans getting in front of any relatively heavy rolling object will simply get you run over or flung into the air with relatively little loss of speed for the object.

Examples (links to things getting hit by cars, don't click if that sort of thing bothers you):

On the highway

At a race (relatively low speed)

This guy makes a pretty good landing of it

This one, not so much

A kick in the face adds insult to injury

A horse doesn't even slow it down

Not even Captain America can do it

And I think a Toronto trolley car is probably an order of magnitude heavier that the heaviest of those vehicles.

So yeah, I think you've got a flawed premise. Not even at slow speeds do people get wedged between cars and the road to somehow slow it down by friction. They get flung or run over.

And as far as getting in front of a moving object and trying to push on it to gradually slow it down, that don't work either. I was unloading a van last week with some supplies for an offsite bar. Loaded up a big flat hand truck with a couple kegs and a bunch of boxes of stuff. Probably weighed in around five or six hundred pounds. I got in front of it and angled it down the very slight decline towards the freight elevator. I'm a pretty big and strong guy, and there was no way I could have stopped that thing once it got rolling. The best I was able to do, pushing on it with all my strength was to slow it down to a controllable speed. Stopping was right out of the question, and I'm pretty sure that if I had stumbled it would have rolled right over me.
posted by Jawn at 12:19 PM on October 6, 2012

Response by poster: I finished the video on the matter.
posted by RobotHero at 8:26 PM on March 6, 2013

Response by poster: To support the idea of not getting flung up in the air, I submit this cringe-inducing video. (He does survive) I think if the car hood hits you below your centre of gravity, you can get flung up in the air, but something flat on the front like a bus or trolley will knock you down.

I have less support for the idea that the trolley won't just roll right over you, but honestly I just didn't want to deal with shear strength of flesh or whatever I would need to address in that circumstance.
posted by RobotHero at 8:39 PM on March 6, 2013

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