SPOILER WARNING!!!

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From the Book

They broke into a sprint, Will pulling in front immediately. As he neared the edge, he heard the stairwell door burst open. Two men in black backed onto the roof, firing back down the stairwell. They turned as suit man screamed, swinging their guns around and firing.

Three steps left.

Two.

One.

The Problem

During the escape from the police station, Will and Besra are forced to choose between certain death at the hands of the terrorists, and jumping off the building. To one side there’s a deep diving pool at the end of a 25 metre lap pool. To the other, the awnings on top of a multi-story car park that might break their fall. They must decide quickly which fall is least likely to kill them before the terrorists storm the rooftop and kill them anyway.

The Information

The main difference between the two options is the distance the boys will fall and the distance they have to cover sideways to make it to the “safety” of landing on the awning or landing in the deep water of the diving pool (landing in 1.5 metres of the normal lap pool will likely kill them as they hit the bottom).

Several cues from the environment help them estimate where they are.

Their Own Building

The boys are on the 20th floor of the police station building. A typical story is about 3 metres tall, so they’re 60 metres above the ground.

The Carpark

  • The car park is separated from their building by the width of 7 car spaces – a typical car space is about 2.5 metres wide, so that tells us that the car park awning is at least 17.5 metres away.
  • The top of the car park is ten stories lower than their rooftop – so it’s about a 30 metre fall from the rooftop to the awning (not taking into account the height of the awning).

The Hotel Pool

  • The hotel diving pool is separated from their building by the length of a 25 metre swimming pool.
  • The pool is 15 stories lower than their rooftop – at 3 metres a story, that’s a drop of 45 metres.

It’s clear from this information that the hotel pool involves a longer fall (so more time to make it further sideways), but also a greater distance to cover sideways in order to make it to potential safety.

The Calculation

Will needs to work out whether they’re more likely to make it across the gap and land safely in the diving pool or on the rooftop awning. He can work this out in two steps:

Step 1: Calculate how long he’ll fall for, before hitting the diving pool / awning

Step 2: Calculate what sideways speed he needs to make it across the distance in this time

2d_fall_options

Falling from the rooftop to the pool is a vertical distance of 45 metres. When something falls, it accelerates, meaning that it falls faster and faster over time. After 1 second of falling, an object is falling at about 9.81 metres per second (m/s) downwards (that’s about 35 km/hr).  After 2 seconds of falling, an object is travelling at about 19.82 m/s downwards.

There’s a formula for calculating how long it takes for an object to fall a certain distance:

eq1

g is gravity, and is 9.81 m/s/s

h is the height the object falls, measured in metres

If we put in the fall height for the pool, we get:

eq2

So Will has 3.03 seconds to cross 25 metres. That means he must get up to a speed of 25 ÷ 3.03 = 8.25 m/s, or about 30 km/hr. For reference, Usain Bolt reaches a maximum speed of about 44 km/hr.

What about for the other option – jumping onto the awning?

Using the same formula but with a fall height of 30 metres:

eq3

So Will has 2.47 seconds to cross 17.5 metres. That means he must get up to a speed of 17.5 ÷ 2.47 = 7.09 m/s, or about 26 km/hr.

So the running speed requirement for making it to the awning is less than for making it to the diving pool – which is why Will chooses to go for the car park rooftop in the book.

Assumptions / Simplifications

There are a host of other factors that determine survivability, such as the effects on the body of hitting the water and of course of hitting the awning – too strong/rigid and the awning does no good, too weak and the awning might as well not be there. The extra half second of falling for the pool option means they will be falling faster at impact, which may or may not be significant.

Humans need a fair distance to get up to full sprinting speed – most sprinters in the 100 metre sprint don’t hit maximum speed until more than halfway through the race (see graph below):

100m_sprint

The calculations also ignore the effects of air resistance, which is a reasonable assumption when speeds are low.