Projectile trajectories

By Marcus Wilson 01/07/2011 7

Just a quick entry since I haven’t done one this week.  Since I got back from holiday (which was extended by a day by that pesky ash cloud) I’ve been busy marking exams and then helping with our major schools’ publicity event of the year, the Osborne Physics and Engineering Lectures. This year we’ve had some particularly loud and exciting demonstrations, including one on rockets by Steve Chrystall – featuring water rockets hurtling across the sports fields and igniting balloons filled with hydrogen – that kind of thing.

But I’ll leave it till next week before saying much more about those – the blog entry I wanted to do earlier this week but didn’t have time too was something that was much discussed in the UK papers a week or so ago. If you’ve not lived in the UK, you might not understand the obsession that grips the country every June. Without fail, the whole country becomes interested in tennis for two weeks, as Wimbledon fever takes hold. Now, there are two inevitabilities about Wimbledon. First, is that Britain will not have a champion, no matter how much we hope for one (a bit like the inevitability of NZ not winning the rugby world cup). Second, is that it will rain. 

The first thing is going to be hard to fix, but some of the problem of the second has now been alleviated by putting a retractable roof on centre court. This was much in use last week. Now, the scientific discussion this prompted was about how the closed roof changes the nature of the tennis match. I’m no tennis expert (except for those two weeks of Wimbledon) but apparently the way the top players play their game will be different with the roof closed than the roof open, and that is because the game takes on more of a clay-court nature – the ball loses more speed with its bounce and bounces a touch higher.  But why does this happen?

There are two possible mechanisms being talked about, both drawing from the fact that there is increased humidity under the closed roof (the water vapour evaporating from the ground and generated by the thousands of spectators can’t blow away). The first mechanism suggests that the tennis ball simply absorbs moisture and gets heavier and its coefficient of restitution (a measure of how much energy it loses on bouncing) changes. This results in a change to the ball trajectory. The second possible mechanism is that, under denser air, the effects of spin are more pronounced. A spinning ball in flight will experience a bending force on it (the magnus effect). This is what causes the hook or slice on a badly hit golf ball, or, in the case of tennis, the rapid dip in the trajectory due to top spin. The idea here is that the top spin has a greater effect in the higher humidity, so the ball hits the ground at a steeper angle, so bounces up higher.

Maybe both effects are present at once. If you had access to a large space in which you could control the humidity you could test these theories. Or, perhaps more simply, it should be possible to estimate their effects.

7 Responses to “Projectile trajectories”

  • Here is the density of 99.9% RH air at 20 degrees: 1188.709 g/m^3

    Here is the density of 50% RH air at 20 degrees: 1194.318 g.m^3

    I DO get frustrated with cricket commentators (especially) telling me that humid air has a greater density.

    We must thunk up another theory.

  • Yes Ross – that thought crossed my mind when writing – without looking anything up I know that the molecular mass of nitrogen is pretty well 28 g/mol, that of oxygen about 32 g/mol, but that of water (vapour) is about 18 g/mol – i.e. at the same temperature and pressure humid air will have less mass than dry air.

    However, if you are in an enclosed space (like centre-court at Wimbledon with the roof down), what happens to the temperature and pressure? Just thinking logically, if the space were completely enclosed, then any evaporation of water will put more molecules into the air, and so the density will go up. However, we don’t have a completely enclosed space (or the crowd would eventually suffocate). How does this change things? Not sure.

  • One thought could be the evaporation from the ground surface is reduced with higher humidity – and less irradienace form the sun/sky. This would mean that the surface becomes a weee bit more moist as the moisture moves from the subsoil. With the roof off I suspect the ground may have a harder surface due to being dried. The higher bounce may be the increased friction between the ball and the grass. This would cause greater “slow down” of the ball when it bounces. Top spun balls would then jump higher thats for sure. Sticky wickets being good for spin bowling come to mind.

    • Interesting. So it might well be that the ball is ‘fluffing up’ as it picks up moisture and has increased air resistance. Explains the tennis issue. Now, what about the cricket ball. That doesn’t fluff-up, surely?

  • Well 2 out of 3 ain’t bad. The fluffy one is interesting. There is no doubt the Great Selection Debate that goes on between player and ball boy is all about finding the least fluffy ball. They go faster. I heard that for the second serve balls they pick the fuffiest for the greater spin.

    So the idea that high humidity effectively expands the ball is pretty convincing.

  • It might be convincing, but is it correct? Give me a humidity-controlled room, Hawkeye system, a collection of tennis balls, a moderately good tennis player and a summer student and I’ll find out…

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