**Galileo: Throwing cannonballs off The Leaning Tower of Pisa **

This post is 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.

Galileo Galilei was an Italian mathematician and astronomer who (reputedly) conducted experiments from the top of the Tower of Pisa. He dropped various objects from in order to measure how long it took for them to reach the bottom, coming to the remarkable conclusion that the objects’ weight did not affect the speed at which it fell. But does that really mean that a feather and a cannonball would fall at the same speed? Well, yes – as long as they were dropped in a vacuum. Let’s have a look at how we can prove that.

**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

To understand where this equation comes from we note that Newton second law (Force = mass x acceleration) is

F = ma

The LHS of our equation (p_{s}gV – p_{a}gV – F_{D}) represents the forces acting on the object and the RHS (p_{s}Va) represents mass x acceleration.

**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

Next, we can note that in a vacuum F_{D} (the drag force) will be 0 – as there is no air resistance. Therefore this can also be ignored to get:

p_{s}gV ≈ p_{s}Va

g ≈ a

But we have a = dU/dt where U = velocity, therefore,

g ≈ a

g ≈ dU/dt

g dt ≈ dU

and integrating both sides will give:

gt ≈ U

Therefore the velocity (U) of the falling object in a vacuum is only dependent on time and the gravitational force. In other words it is independent of the object’s mass. Amazing!

This might be difficult to believe – as it is quite unintuitive. So if you’re not convinced you can watch the video below in which Brian Cox tests this out in the world’s largest vacuum chamber.

If you liked this post you might also like:

War maths – how modeling projectiles plays an essential part in waging wars.

**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.

## 4 comments

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October 28, 2015 at 10:33 am

Matthew WrightI believe this was also demonstrated on the Moon by one of the later Apollo missions where the astronaut dropped a hammer and a feather.

October 29, 2015 at 5:49 am

Dan PearcyVery nice – love the simplicity of the derivation but the power in the final result. Thanks for posting.

October 29, 2015 at 6:56 am

Ibmathsresources.comThanks for the comments – I have added the video about the moon experiment.

Yes, I was quite surprised that it was so simple to show! I might try and look at some other basic Newtonian laws and what we can derive from them. Next up, parachute jumps!

June 28, 2017 at 12:35 pm

sigoldberg1See also the (to me) far more significant thought exp. of Galileo referenced at https://en.m.wikipedia.org/wiki/Galileo%27s_Leaning_Tower_of_Pisa_experiment.

“Galileo arrived at his hypothesis by a famous thought experiment outlined in his book On Motion.[12] Imagine two objects, one light and one heavier than the other one, are connected to each other by a string. Drop this system of objects from the top of a tower. If we assume heavier objects do indeed fall faster than lighter ones (and conversely, lighter objects fall slower), the string will soon pull taut as the lighter object retards the fall of the heavier object. But the system considered as a whole is heavier than the heavy object alone, and therefore should fall faster. This contradiction leads one to conclude the assumption is false.”. For further clarity, the reader might imagine a large ball and a small ball made out of the same material, connected by a wire made out of the same material. Now let the connecting wire get thinner and thinner. At what point does the velocity of falling decrease because one object becomes two?