Trusting your life to your own physics calculations

Alison Campbell alerted me to the following: Physicist Andreas Wahl shoots himself with a gun underwater – and proves a point about drag force. 

For the record – I won't be repeating this. Physics or no physics, the guy is crazy. 

BUT, what I have done, is a quick post-hoc analysis from the safety of my own office. There's a little bit of maths involved, but the gist of it is this. The drag force on an object (in turbulent conditions – which this most certainly is), is given by the equation c rho A v2 where c is the 'drag coefficient', rho is the density of the fluid in which the object moves, A its cross-sectional area and v its speed. If we equate this to the objects mass times acceleration (Newton's second law) we get an expression for the acceleration of the object in terms of some physical parameters. Solving the equation (integrate it!) gives an exponential decay relationship between the velocity of the object and the distance it travels. Thismeans there's a characteristic length-scale, d, given by:

d = rho_bullet x b / (c rho_liquid)  

where b is the length of the bullet.  Broadly speaking, d gives you an indication over what distance the speed will decay over. We can now stuff some numbers in. Let's assume the density of the bullet is about five times the density of the water (note how it's only the ratio of the two that matters) and that the bullet is about 2 cm long. The drag coefficient will be quite low, given its a streamlined object; say about 0.1. That gives a distance scale of around a metre. How far does the bullet travel? From the video, I'd say something around a metre. 

So, what's the difference between water and air? It's the density of the fluid. Air has a density around 1 kg per metre cubed (rather than water's 1000 kg per metre cubed). Fire the bullet in air, and the length scale goes to one thousand metres (1 km). I'm not a gun expert, but that figure seems about right. 

I also think he was very wise to have his head out of the water. Sound travels rather well underwater.

And it's very easy to say all that without a gun pointing at your chest. 


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