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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?