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## Fun with Wind-Resistance (Part 3) – Optimal mass

Intuitively, there is an optimal mass for a ball being thrown. If it’s too heavy then we won’t be able to give it a large initial speed. Too light, and it will be slowed down very quickly by air resistance. A shot is too heavy, a tennis ball too light.

To calculate the optimal mass for a projectile, we need to have a model for how a thrower accelerates the ball before release. I will make what is perhaps the simplest assumption: the force applied by the throwers arm and the distance over which that force is applied are held constant. This is equivalent to assuming that the thrower will impart a fixed amount of kinetic energy (K) to the ball. Then, the initial speed (v) of the ball varies with the mass (m) as,

$v = \sqrt{\frac{2 K}{m}}$

K will be fixed using the fiducial case of a cricket ball thrown with initial velocity of 120, 140 and 160 km/h. As before, the launch angle is chosen to maximise the range of the throw for a 1.8m tall thrower.

The plot shows that, as expected, there is a mass which maximises the range of the throw. It is quite close to the actual mass of a cricket ball (0.16 kg, dashed vertical line) and a baseball (0.145 kg), which is a satisfying result. The optimal mass increases slightly with the force applied by the thrower (i.e. the fiducial initial velocity $v_0$).

Next time: how much easier is it to hit a six (or a home run) at higher altitudes?

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### One Response

1. Now to have the philosophical argument. The ‘design’ hypothesis is that this optimal mass for the cricket ball was carefully calculated by repeated trial and error, the alternative hypothesis is that it is an accident. There are no prizes for suggesting that there are a multitude of cricketing universes, each with a different cricket ball mass, and we have naturally evolved in the optimal universe because cricket is essential to the highest form of life! : )