Giving it your best shot

By Burkard Polster and Marty Ross

The Age, 4 August 2008

How do you become a great shot putter? Work hard and eat your Wheaties. Then you might follow in the footsteps of Justin Anlezark, representing Australia in three Olympic Games. And, maybe applying a little mathematics will help get you there.

Shot putting is a great application of the mathematics of projectiles. We launch the iron shot with a vertical speed V and a horizontal speed H. Then the Earth’s gravitational acceleration g determines when and, most importantly for a shot putter, where the shot lands.

If we launch from the ground then the vertical speed V determines the time in flight to be 2V/g. Multiplying this time by the horizontal speed H, we find that the shot lands away a distance 2HV/g. This calculation ignores air resistance, but the high density of the 16 pound shot means that the resistance has little effect.

How do we make the distance traveled as large as possible? Obviously, by throwing harder (hence the Wheaties). But given we can only throw at some maximum total speed, our task is to find the optimal launch angle. Most of us have experimented with this: just recall trying to douse your annoying brother with the garden hose, and struggling to get the water stream to reach as far as possible.

Forming a rectangle with base H and height V, the total speed T is the diagonal. So, making the distance 2HV/g as large as possible amounts to adjusting the launch angle to make the area HV of this rectangle as large as possible.

You are probably not surprised that, for fixed diagonal T, the rectangle of largest area is the square. That is, we ensure H=V, giving an optimal launch angle of 45°. Contact us for a beautiful proof of this. (Note also, if your brother has commandeered the hose then your strategy is to convince him that the best angle is 90°).

These calculations are not completely realistic. First of all, an actual shot putter will launch the shot about 2 meters above the ground. The consequence is that it is better to launch at a slightly shallower angle. At the speed shot putters throw, the optimal launch angle is about 42°.

Secondly, shot put is not just mathematics. There are also biomechanical considerations. It turns out that shot putters can throw harder at shallower angles. So, there is a trade-off, with most shot putters throwing around 35°.

In fact, Justin Anlezark is unusual in that he throws close to the mathematical optimum angle of 42°. And he throws well enough to be a serious medal chance in Beijing. But does Justin use the maths to analyse his technique? “Nah. If you throw 20 meters at 42°, or you throw 20 meters at 30°, you just think ‘Hey, I threw 20 meters!’ ”