Here is a question I’ve been mulling over for a few days since I heard a cricket commentator raise it during the recent West Indies – South Africa Twenty20 match. How high do you need to hit a cricket ball in order for it to reach terminal velocity on its way down? – in other words, beyond what height does the height of the ball make no difference to the speed at which it hits the hands of the unlucky fielder underneath?
The commentator’s question raised a flurry of email responses, which were read out during the course of the match, some of which sounded somewhat bizarre to me.
Anyway, here is my take on it. A cricket ball is subject to air resistance which scales as its velocity squared. The relevant formula also has factors for the density of air, the ball’s radius squared, pi, and the ‘drag coefficient’ (divided by two). Putting some values in for a cricket ball, that gives me a force (in Newtons) of about 0.003 times the velocity squared (velocity in metres per second). At terminal velocity, this is equal to the weight of the ball (so drag = weight, and Newton’s first law says a balanced force gives no change in velocity, that is, terminal velocity has been reached.) The weight is about 1.6 Newtons, so equating the two means velocity is about 22 metres per second, or around 80 km an hour. Fancy taking that catch?
Now for the height. The acceleration due to gravity is about 10 metres per second, per second, which means that to get to 22 metres per second velocity we’d need a height of about 25 metres, and the ball would take about 2.2 seconds to fall. That’s not really all that high.
This is a very very rough estimate, but it gives you some idea. Of course, knowing this makes no difference whatsover to my chances of actually holding on to the catch.