**How to Design a Parachute**

This post is also inspired by the excellent book by Robert Banks – Towing Icebergs. This book would make a great investment if you want some novel ideas for a maths investigation.

The challenge is to design a parachute with a big enough area to make sure that someone can land safely on the ground. How can we go about doing this? Let’s start (as in the last post) with some Newtonian maths.

**Newton’s Laws:**

For an object falling through the air we have:

p_{s}gV – p_{a}gV – F_{D} = p_{s}Va

p_{s} = The density of the falling object

p_{a} = The density of the air it’s falling in

F_{D} = The drag force

g = The gravitational force

V = The volume of the falling object

a = The acceleration of the falling object

**Time to simplify things**

Things look a little complicated at the moment – luckily we can make our lives easier through a little simplification. p_{a} will be many magnitudes smaller than than p_{s} – as the density of air is much smaller than the density of objects like cannonballs. Therefore we ignore this part of the equation, giving an approximate equation:

p_{s}gV – F_{D} ≈ p_{s}Va

We now rewrite things to make it easier to substitute values in later.

p_{s}V = m, where m = mass of an object (as density x volume = mass)

This gives:

mg – F_{D} ≈ ma

and as mg = W (mass x gravitation force = weight) we can rewrite this as:

W – F_{D} ≈ (w/g)a

Now, the key information to know when looking at a parachute design is the *terminal velocity *that will be reached when the parachute is open – that means the maximum velocity that a parachutist will potentially be hitting the ground traveling.

Now, when a person is traveling at terminal velocity their acceleration is 0, so we can set a = 0 in the equation above to give:

W – F_{D} = 0

Now we need an equation for F_{D} (the drag force).

F_{D} = 0.5p_{a}C_{D}AU^{2}

where

p_{a} = density of the air

C_{D} = the drag coefficient

A = area of parachute

U = velocity

So

when the parachutist is traveling at their terminal velocity with the parachute open we have:

W – F_{D} = 0

W = 0.5p_{a}C_{D}AU^{2}

OK, nearly there. Next thing to consider is what is the maximum velocity we want someone to be traveling when they hit the ground. This is advised to be around 5 m/s – similar to jumping from a 2 metre ladder. Much more than this and you would risk breaking a bone (or worse!)

So we are finally ready to solve our equation. We want to find what value of A (the area of the parachute) will make us land safely.

We have:

p_{a} = 0.6kg/m^{3} (approximate density of air at 3000m)

C_{D} = 1.40 (a calculated drag coefficient for an open parachute)

U = velocity = 5m/s (this is the maximum velocity we want to want to avoid injury)

W = 100kg (we will have this as the combined weight of the parachutist and the parachute)

So,

W = 0.5p_{a}C_{D}AU^{2}

100 = 0.5(0.6)(1.40)A(5)^{2}

A = 9.5m^{2}

So if we had a circular parachute with radius 1.7m it should slow us down sufficiently for us to land safely.

**IB Revision**

If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

The Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial. Really useful!

The Practice Exams section takes you to ready made exams on each topic – again with worked solutions. This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.

## 2 comments

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January 20, 2017 at 1:50 pm

Maria KingsleyCould you please do my maths exploration?

January 20, 2017 at 1:51 pm

BenOh, I want you to do mine, also!