Mumbai Locals...

Well Mumbai locals are the life line of the city. But ever wondered how many people can one local train carry? Here I try to estimate the carrying capacity of the local train.



We first want to make an order of magnitude guess for the carrying capacity of the
local train. First let us take the dimensions of one coach of the train.
Let us take the width of the coach to be ~ 3 m or 10 ft. We consider the length of the coach to be
of the order of ~50 ft. Then the floor area that we have in each coach is about 500
sq. ft. We neglect the actual seating arrangement in the local, and consider the
floor area only. We make an assumption that all the people are standing in the coach to
get an upper limit on the carrying capacity of the coach. The passengers are standing
as close to each other as possible. Now we make an estimate of how much area one
person requires to stand. One person would require about 1 sq. ft. area to stand.
Thus in a coach of about 500 sq. ft, about 500 people can stand. Actually there are
9 coaches, and their configuration is as follows. In the Central Railways , a 3-coach
unit is classified as 76, 70, or 72, where 76 is the leading motor coach, 70 is the motor
coach with a pantograph, and 72 is the trailer coach. So a nine-coach train has three
units in the following sequence (for the details and lot of other interesting information about Indian Railways visit here):

(76 -70 - 72)(72 - 70 - 76)(72 -70 - 76)

So in out of 9 coaches some space is lost to the motor coach [3 nos.], the driver
coach [2 nos.] and the eeffective area of the train is reduced. The motor coach has an
area of about 10 ft. and the driver coach of about 5 ft, so about 40 ft is reduced. So
the eeffective number of coaches are 8. Since each coach can hold about 500 people,
8 eeffective coaches will have about 4000 people. We have given about 1 sq. ft. for
one person to stand, but in reality especially in the peak hours the rush is much more
than that, so this estimate will have to be increased. We consider that about 1.5
people can stand in 1 square foot of area. Also the presence of the seats and partitions
in the coaches will reduce the eeffective area usable for standing so we assume that
about 10 % of the entire area is lost in furniture. So the number of people in one coach
450*1:5 = 675. So that in 8 coaches 675*8 = 5400 people can stand. But since not
all people can stand we also have to make a correction for this. About 100 people can
sit in a coach, who effectively take about 2 sqf ft. So about about 150 sq. ft. is taken
by them. So out of the 450 sq we are left with 300 sq ft, so eeffectively 300*1:5 = 450
people are standing. So the total number of people per coach is 450 + 100 = 550. So
that total number of people per train is 550* 8 = 4400. The figures that we get from
Wikipedia show that about 4500-5000 people travel in the local trains during the
peak hours.

So our guess is near about correct!!

This method of analysis is known as solving problem the Fermi way and the problems are Fermi problems. Named after the 20th century physicist Enrico Fermi, such problems typically involve making justified guesses about quantities that seem impossible to compute given limited available information. Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name.