# Base Jumping

## The Scenario

You are the world’s best base jumper and wing suit flyer, famous for your daring in jumping off anything and everything tall. However, behind all the show there’s some careful calculations that are done, to make sure you are famous for staying alive, rather than the grimmer alternative.

## Math Scenario 1: When to Pull the Chute (No Air Resistance)

For your latest jump, you’ve received permission to jump off the Burj Khalifa, the world’s tallest building located in Dubai in the United Arab Emirates.

You’ll be jumping from the top of the spire, which is 830 metres above the ground, and need to work out when to pull your chute.

Your team tells you that the latest you can pull your chute is when you are 400 metres above the ground. How long after jumping will this be?

*Allowable free fall distance = jump height – chute height*

*Allowable free fall distance = 830 – 400*

*Allowable free fall distance = 430 m*

You will fall, accelerating at the rate of gravity. In reality, air resistance would mean you wouldn’t fall quite as quickly, but ignoring air resistance will give you a safer figure to work off.

*Fall distance = 0.5 * g * t^2*

*t^2 = 2 * fall distance / g*

*t = sqrt(2 * fall distance / g)*

Note the mathematical answer should be plus-minus the square root, but the positive answer is the one that makes physical sense in this case.

*t = sqrt(2 * 430 / 9.81)*

*t = 9.363 seconds*

The team also needs to calibrate the chute for the maximum speed you’ll be travelling at chute deployment, in km/hr:

*max speed at chute deployment = fall time * gravity*

*max speed at chute deployment = 9.363 * 9.81*

*max speed at chute deployment = 91.85 m/s*

*max speed at chute deployment = 91.85 m/s * 0.001 m/km * 3600 s/hr*

*max speed at chute deployment = 330.66 km/hr*

However, this speed is faster than the maximum velocity a free fall human will reach due to *air resistance*, so we can perform a more sophisticated calculation using air resistance.

## Math Scenario 2: Free Fall With Air Resistance

The formula for how far an object will fall when considering air resistance is:

where *v_infinity* is the terminal velocity of the object, and *y* is the starting position. *ln* is the natural log, and *cosh* is the hyperbolic cosine.

In the last question, we worked out that we could free fall 430 metres in 9.363 seconds. Let’s see how far we’d fall during that time with a consideration of air resistance.

We can use a terminal velocity of 56 m/s for a human.

y = 0 – (56^2 / 9.81) * ln.cosh(9.81 * 9.363 / 56)

y = -314.6 m

So air resistance makes a sizeable difference – instead of falling 430 metres, with air resistance the free falling base jumper will have only fallen 314.6 metres.

## Math Scenario 3: Viable Slope for Proximity Flying

Proximity flying is an even riskier version of wingsuit flying where participants deliberately fly very close to the faces and ridges of mountains and cliffs. This variant of the sports is even less tolerant of miscalculations, because there is so little margin for error.

You’ve scouted out a new proximity flying location that looks promising, but you’re a little concerned about a rock outcrop near the bottom of the route. You need to calculate whether the route is possible, both with and without going over the rocky outcrop.

You’ve taken the following readings for the bottom section of the route (shown in the picture above):

**Top of section:** altitude 1400 m

**Bottom of hill section:** altitude 700 m, horizontal distance 1300 m

**Rock at bottom of hill:** altitude 770 m

By this part of the jump, you are gliding at a constant 2:1 ratio – for every 2 units of distance you travel sideways or horizontally, you fall 1 unit of distance.

First we can check whether the route itself (minus rock) is possible. We’re looking for a horizontal distance to vertical fall distance ratio of less than 2:1:

*Fall ratio = horizontal distance / altitude change*

*Fall ratio = 1300 / (1400 – 700)*

*Fall ratio = 1.857*

So without the rock it should be okay. What about with the rock?

*Fall ratio = horizontal distance / altitude change*

*Fall ratio = 1300 / (1400 – 770)*

*Fall ratio = 2.063*

Clearing the rock requires a better glide ratio than you can pull off – so you’d better remove it from your plans!

## Real-Life Example: Burj Khalifa Record Base Jump

In 2014 a pair of skydivers (and their camera person!) set a new base jump record by jumping off the Burj Khalifa. To get the record, they had an *extra* platform added to the top of the building which added about 15 metres of extra height.

From the video, it looks like they had several seconds of free fall before they opened their chutes. Normally base jumps from buildings involve the jumper pulling their chute as soon as they jump, simply because there isn’t enough height to play with.

But with the Burj Khalifa being twice as tall as most tall buildings, it offers base jumpers a unique (and presumably highly desirable) experience.

You can see their amazing jump here: