I like this question. It caused me to think more carefully to come up with a plausible explanation.
Your intuition and experience have lead you to the conclusion that important parameter is the velocity of the ball relative to the wind.
It?s always a good sign when the model we get from our physics equations agrees with our intuition and experience. So let?s try to figure some of this out. Or better yet, let?s take advantage of the efforts of others. I leave the complicated stuff as a sort of appendix to the much simpler answer.
A big part of fluid mechanics or dynamics, is the motion of objects through a fluid. For your question, this is the ball moving through the air. Air resistance, or drag, is a very complicated effect to model. In fact, fluid dynamics often involves detailed mathematics which can cloud the understanding. In just about all cases, the push from the wind is much more important than the drag! Usually by several orders of magnitude ( How weather affects baseball and MadSci answers below).
So before I venture forth into the land of fluid dynamics to consider drag, let?s gather what we already know and state a couple assumption.
Let?s assume the ball moving through the air does not generate lift like a spin golf ball or a discus thrown into a head wind. The ball would have to be spinning extremely quickly to generate lift. Is it possible that a ball moving faster than the wind will generate some sort of vacuum pull to slow it down? The answer is not enough to be any real noticeable effect. This happens with airplanes, but does not happen with balls. Typically the air passing over a ball separates and becomes turbulent which reduces its effect to create drag.
First, that with the same initial launch angle and all other things except initial speed being equal, the faster thrown ball will go farther. In other words, for a given wind in any direction, the faster thrown ball will go farther. Also it can be shown that the maximum range (horizontal distance traveled) for the case of zero air resistance is 45 degrees. I refer you to a high school physics text for the derivation of the range equation (projectile motion).
A tail wind is always going to help, no matter how much faster the ball is traveling relative to it.
The way to think about this is to envision an airplane traveling with a tail wind. Clearly, the airplane is faster than the wind. Even if we shut off the engines it will continue to glide faster than the wind for some time. This is the addition of velocity vectors (Galilean relativity). This is perfectly valid at speeds of everyday experience (i.e. not anywhere close to the speed of light).
Let?s now consider the air resistance.
There are three forces then acting on the ball:
- Gravity always acting down.
- Air resistance, or drag which opposes the instantaneous direction of motion (opposed to the ball?s velocity).
- The wind always pushing the ball in the direction of the wind.
Let?s look at what has to happen (very simplistically) to get the ball to go the same distance with a tail wind and a head wind. We want to consider the three velocities in our equations: the wind velocity, the velocity of the ball in the fluid, and the velocity of the ball relative to the ground ? which is the resultant of the wind velocity and the ball velocity in the wind. This may seem like a backward way to look at the problem at first, yet it really is not. It is the velocity relative to the ground that determines how far the ball travel.
Without going into significant detail here, the force of the wind on the ball is of the some form as the force of air drag (proportional to speed squared).
Below are the force diagrams of the ball in flight against and with the wind. In the figures the ball is moving upward. The vector U is the speed of the wind relative to the ground (assuming horizontal wind).
These force diagrams illustrate the effect of the wind and drag, as well as gravity on the ball. The wind against the ball will always retard its motion. The vector
V in fluid is the actual magnitude and direction with which the ball is moving in the air at the given instant. The Vector V is the velocity of the ball relative to the ground. These diagrams have been set up such that V is the same, demonstrating that the ball must be thrown at a more shallow angle and harder (i.e. more speed) to get the same horizontal displacement relative to the ground. The direction of the drag force is opposed to V in fluid, but the ball travels at a velocity vector V relative to the ground. For a baseball traveling along at 50 to 90 mph, the drag force is about 0.5 to 1.1 times the weight of the ball.
We all get the intuitive result that the ball thrown hard has more drag on it. We should note that I call these instantaneous forces. This is a snapshot of the ball at a particular time during flight. For the case of the ball traveling at the speed of the wind, where Vx ? U = 0, there still could be a vertical component to the velocity. In this case the drag force will oppose the only vertical motion. The velocity angle q will be 90 degrees. So in this picture the ball is just moving up in down in the relative to the fluid, yet moving along a horizontal speed U relative to the ground.
The general conclusions are:
To get a ball to go farther into a head wind, it must be thrown harder and lower relative to no wind or a tail wind.
A ball moving at the speed of the horizontal wind will experience no drag in the horizontal direction.
A tail wind is always going to help, no matter how much faster the ball is traveling relative to it (stated earlier).
Sincerely,
Tom ?Big Ol? Wind Bag? Cull
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Appendix
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For completeness, I include some background and quick summary of ideas found on the web and in text books.
I found a very detailed and technical web page about numerical solutions to the differential equations of a thrown baseball and drag.
Flight of a Baseball with Drag. The take-away from this web page is that, as expected, a ball with less drag travels farther (Figure 3). The other thing we learn (trusting their results and our intuition and experience) is that as drag increases we need to throw the ball on a flatter trajectory (less than 45 degrees) because the drag is in effect for the entire flight path of the ball (Figure 4).
For even more information, and in some ways less mathematical, please refer to some previous questions and answers on air resistance can be found at MAD.SCI Search Form using the keywords : air resistance ball.
Here a few I found responses related to your question.
Re: Will 2 falling objects of diiferent mass hit at the same time?
Re: What forces affect a ball bearing in jelly?
Re : Do all objects fall at the same speed?
Re: What would allow to solid lead balls to hit the ground at different time?
Re: How do temp, press, humidity affect the distance traveled by golf balls?
Re: Does strength or leverage determine the distance one can throw a baseball?
Re: Two gravity questions driving me nuts! Please!
(one of my previous answers with lots of links to other queries also explains regimes of speed dependence).
Please note that there seems to be a bit of inconsistency (again fluid mechanics is complicated and easy for those less familiar with it to make mistakes). One answer considers air resistance as proportional to the velocity. In general, the air resistance of a ball is proportional to square of speed relative to the fluid it is moving in (i.e. air) and opposed to the direction of motion. Only in the case of extremely slow motion relative to the viscosity (measured as the Reynolds number) does the resistance come into the equations proportional to the speed (like a ball in thick oil). At typical sports ball speeds and normal air conditions, we can do pretty well using the square of speed. The problem is that if the drag is proportional to the velocity, the problem can be solved (complicated, but it can be done ? college level mechanics).