Basic Model for Optimal Javelin Throw

During the recent telecast of the Olympics, one of the things that was very counter-intuitive to me was that the javelin throw contestants were throwing at an angle of around 30–40 degrees, as opposed to the conventional wisdom of launching projectiles at 45 degrees for maximum range.

Renowned Indian Javelin thrower Neeraj Chopra in action (Erik van Leeuwen, attribution: Erik van Leeuwen (bron: Wikipedia)., GFDL <http://www.gnu.org/copyleft/fdl.html>, via Wikimedia Commons)

The initial guess was that the difference could be attributed to the height at launch. Using basic kinematics, we can write down the expressions for the distance travelled and the height at which the javelin is launched. These are given by:

Here u represents the launch velocity, θ represents the javelin angle with respect to the ground and h is the height at launch.

The latter equation is a quadratic and only the positive root is considered. This gives us a straightforward objective function that needs to be maximized as a function of the angle. It can be written as:

This can be easily optimized numerically and a short Python snippet is used for the same. Usually, javelins are launched at a velocity of ~100 km/hr and a range around this value is used for the calculation.

Figure 1: Optimal angle of javelin versus release velocity and height around typical human ranges

It can be seen that optimal values for the angle are around 44 degrees, which is close to 45, but not the same. From this, it can be concluded that simple kinematic explanations might not be sufficient to elaborate on why athletes use angles that are more in the mid-30s.

Javelins are non-uniform and air resistance also likely plays an important role in not just the range, but how the javelin tips over its trajectory. Those are topics that are worth considering in a more complicated model to predict the optimal range.