Part of University of Groningen

# Surprising cannon

Episode 3

You don’t always need a big laboratory for exciting experiments... However, a bit of space could come in handy if you’re going to build this vacuum cleaner cannon. Caution! This episode contains mathematics!

‘A mathematician is a blind man looking for a non-existent black cat in a dark room.’ I didn’t dream that up; it’s a quote attributed to Charles Darwin. Darwin forgot that a scientist without mathematical knowledge is blind too. Without mathematics you couldn’t even find a Bengal tiger with a siren on its head in broad daylight – so to speak.

First things first. Before you can start doing any mathematics, you’ll need to convert the vacuum cleaner into a cannon.You will need a 40mm PVC pipe of about one to one and a half metres long.You will also need a T section that fits on the pipe, preferably with a 45 degree angle.You can buy the pipes at a DIY store, and while you’re there you can also get a roll of wide painter’s tape or, even better, duct tape.

Go to a supermarket and buy a dishcloth and two or three Kinder Surprise eggs. Throw or give away the surprise in the egg and eat the chocolate:all you need is the plastic capsule containing the surprise, because it’s going to be your cannonball. By the way, you can use the capsules for other experiments too – so don’t throw them out after this one.

Place the T section on one end of the pipe and fix it with tape if necessary. Now cut a two-centimetre wide strip off the dishcloth and wrap this tightly around the vacuum cleaner nozzle. Insert the nozzle into the diagonal of the T section and affix it with tape – make sure you keep the cannon barrel clear.

Now you’re ready to fire. Hold a piece of cardboard against the end of the barrel closest to the diagonal, turn on the vacuum cleaner and hold the plastic egg at the front of the barrel. Two warnings: keep your fingers away from the end of the barrel (ouch!) and don’t aim at windows, lamps or pets.

So why doesn’t the cannonball disappear into the vacuum cleaner? Mass is inert, or slow. This means, to put it simply, that still standing objects don’t like being put into motion and that moving objects don’t like changing direction. Sir Isaac Newton described the principle in his First Law: ‘Every body remains in a state of rest or uniform motion unless it is acted upon by an external unbalanced force’. There is also a practical, non-Newtonian explanation: the plastic egg won’t fit sideways through the pipe!However, the theory holds true.

And now the maths: at what angle do you need to hold the cannon barrel to shoot the egg as far away as possible? The egg’s speed, or velocity (v), is actually made up of two components: the horizontal speed (vx) and the vertical speed (vy). The ratio between the two components is determined by the angle (a) under which the egg is fired: the more vertical your shot, the greater vy will be, while if you fire horizontally vx will be greater.

When the egg leaves the barrel, there is no longer a horizontal force working on it, so vx will remain constant (see the First Law). Gravity works on the egg in the vertical direction, changing vy with -gt, where g is gravitational acceleration (9.8m/s2) and t is the time in seconds.

In less mathematical terms: the number of seconds that the cannonball remains in the air is determined by the vertical speed, and the distance the cannonball travels in time is determined by the horizontal speed. If you combine that with the correlation between vx and vy in a formula, you get:

vxt = vcosa*2(v/g)sina

Apply some simple math (see the website), and the result is an angle of 45 degrees. But you may have already guessed that beforehand.

Author: Ernst Arbouw