## Sunday, November 29, 2015

### When Shopping Could Kill

This year at Macy's,  Black "Friday" began at 6:00PM on Thanksgiving Day. I find this reprehensible. Someone has to work on Thanksgiving so that hundreds of thousands of people can race, climb over, and fight each other for the best savings. People have gotten hurt, even been killed by the rush of shoppers. My goal with this blogpost was to find a case study and look at the kinematics of such a huge crowd of people and the resultant risk of being shoved or trampled in the initial rush.

The example I found was Macy's Flagship store at Herald Square in NYC. According the ABC News, 15,000 people waited on line for the doors to open at 6:00 pm. (http://abcnews.go.com/Business/black-friday-off-roaring-start/story?id=35444641)

I decided to ask, if all 15,000 people sprinted into the store, what would their combined momentum be? Their kinetic energy? What would happen if this horde of people ran into a wall? What would happen to a person who fell under their feet and was trampled?

If the average mass was 75kg and each person ran at 6.7m/s (the average sprint velocity according to ask.com) their combined momentum would be:
p=mv
p=15,000*75kg*6.7m/s
p = 7.54*10^6 kgm/s
Their kinetic energy would be:
KE=.5*m*v^2
KE=15,000*.5*75kg*(6.7m/s)^2
KE = 2.53*10^7 J

If the crowd had to stop, for instance, if the front row hit a wall, how many people would be crushed? We'll assume everyone has rubber soled shoes on, and the floor is made of concrete, The coefficient of kinetic friction between rubber and concrete is .6-.85 according to Wikipedia. We will use .725 If everyone tries to stop by sliding on their feet, it will take:

KEf-KEi=Wnc=Fdcos()
cos()=-1      F=mg
-2.53*10^7 J = .725(15,000*75kg*9.81m/s^2)d
d=3.16m

If each person has a depth of .5m, this means that at least six rows of people will be crushed against the wall before the crowd comes to a halt. From the picture below, it looks as if each row coming down the hall is 15 people wide. That's 90 people.

Of course, crowds don't really move like that, but I also did not take into account the greatest factor that would make the crowd take longer to slow down: the difficulty in communicating the need to stop.

A more realistic situation: A single person falls. It has happened before. If, as we assumed before, the crowd is 15 people wide, it will be 1,000 people long. If the whole crowd stops as before, it will still take 3.16 m to stop the whole crowd, so the individual will already have been trampled by 6 people (if the crowd is very organized, that is). More likely, the crowd doesn't stop, and all 1,000 people run over the poor individual.

The moral? Black Friday is Very Scary and everyone should stay home.