Wikipedia:Reference desk/Archives/Mathematics/2011 June 17

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June 17

How wet will the windshield get?

Let us imagine a car which has to travel a set distance, say 100 meters, between two garages during a rainstorm. The rain is falling horizontally at a constant even rate for the entire period of the trip, say one liter per cubic meter per second. The car's windshield is a flat rectangular surface of, say, two square meters, at some angle, Θ between 0 and 90 degrees normal to the rainfall. The car is travelling at some speed v on a perfectly straight even course between the garages. Of what variables in what degree is the amount of rain that will hit the windshield (during a trip between the garages at any finite positive speed with the windshield at a given angle) a function? Can you provide a formula? Thanks. μηδείς (talk) 03:18, 17 June 2011 (UTC)

"falling horizontally", do you mean falling vertically? "one liter per cubic meter per second", do you mean one liter per square meter per second, or one liter per cubic meter ? If each raindrop falls u meter downwards per second, and the car moves v meter forwards per second, and there is ρ kilograms of water in each cubic meter of air, then there will fall ρ·u kilograms water per second per square meter horizontal surface, and ρ·v kilograms of water per second per square meter vertical surface. The car moves s meters from one garage to the other in t = s/v seconds, so the horizontal surface receives ρ·u·t = ρ·s·u/v kilograms of water per square meter, and the vertical surface receives ρ·v·t = ρ·s kilograms of water per square meter. The surface area of the windshield is A square meters, and the angle of the windshield against the vertical is Θ, so the horizontal area is A·sin(Θ) and the vertical area is A·cos(Θ). So the total amount of water falling on the windshield is

ρ·s·A·(u·sin(Θ)/v+cos(Θ))

kilograms. Bo Jacoby (talk) 09:14, 17 June 2011 (UTC).

This sounds like the perennial problem of what speed should I walk at or run through the rain, see [1] for instance for starters or just search for something like "walking in the rain maths". If you walk long enough you'll get so wet that it'll become impossible to get any wetter so at some stage you almost certainly get drier, now that's the sweet spot ;-) Dmcq (talk) 14:47, 17 June 2011 (UTC)

It occurs to me I should have defined the velocity of the vertically falling rain, rather than just giving the density per second which leaves the falling velocity undefined. Lets say there are a liter of drops per cubic meter and they are falling straight downward at 10 meters per second. I suspect the angle of the windshield makes the question a little more complicated than the question as to whether a person should walk or run in the rain, and Bo Jacoby's equation strikes me as taking into effect the relevance of the angle of the windshield, but I'll have to wait til I have time to sit down and visualize it. μηδείς (talk) 15:57, 17 June 2011 (UTC)

The aerodynamics of the airflow over the windscreen will make a substantial difference, so the calculated result will not be an accurate answer, but there are too many variables to bring aerodynamic considerations into the calculation, and the effect reduces as the size of the raindrops increases. Dbfirs 07:57, 19 June 2011 (UTC)

Dotted variables

I came across some formulae I want to understand, which contain elements with single or double dots or horizontal lines above them. What are those? DirkvdM (talk) 06:21, 17 June 2011 (UTC)

They often denote the first and second derivatives of the variables, respectively, especially in the physical sciences (kinematics, dynamics, etc.). —Anonymous Dissident^Talk 06:55, 17 June 2011 (UTC)

Ah, that gave me somewhere to look. Thanks. The dots are Newton's notation for derivatives.

That still leaves the lines. For example, Newton's second law (F=ma) is written as F = d/dt (mU), where the F and U have lines over them. U stands for speed here, and acceleration is change in speed, so that makes sense to me. But what do the lines mean? DirkvdM (talk) 08:42, 17 June 2011 (UTC)

Lines above a variable (often called bars) often indicate that the variable represents a vector. Force and velocity (not speed) have both magnitude and direction, so they are vectors.--Antendren (talk) 08:55, 17 June 2011 (UTC)

Ok, thanks! Now I can start looking things up and brush up on my secondary school maths (too long ago - I forgot). Oh, and that article says that speed is the magnitude of velocity, so I also get that remark.

The differentiation stuff is slowly starting to come back to me now. But I remember never really getting the vector-stuff. As I recall, I could 'do the sums', but without understanding what I was doing. How should I read ${\vec {F}}$ = d/dt (m${\vec {U}}$)? "The direction of the force is the change in the direction of the impulse"? That doesn't make any sense to me. DirkvdM (talk) 10:08, 17 June 2011 (UTC)

The vector notation is just shorthand for each of the three directions, e.g., ${\vec {F}}=(F_{x},F_{y},F_{z})$. Doing the differentiation does not mix the components (there are other vector operators, which do this, like the cross product and the divergence). Thus, Newtons second law in the form stated is equivalent to

${\begin{aligned}F_{x}&={\frac {d(mu_{x})}{dt}}\\F_{y}&={\frac {d(mu_{y})}{dt}}\\F_{z}&={\frac {d(mu_{z})}{dt}}\\\end{aligned}}$

So the x-component of the resulting force is the time derivative of the x-component of the momentum, and so on. --Slaunger (talk) 10:32, 17 June 2011 (UTC)

Vectors are not just direction, they have both a magnitude and a direction. While you could talk about how the derivative of a vector affects its magnitude and direction separately (the parallel component changes magnitude, the orthogonal component changes direction), it's easiest to consider differentiation in standard components as Slaunger describes.

Philosophically speaking, vector notation is not "just shorthand" for components. Components are just an easy way to deal with them. -- Meni Rosenfeld (talk) 13:31, 17 June 2011 (UTC)

Be careful using the subscript letters, e.g. F_x, etc. This notation is often used to denote the partial derivative with respect to x, i.e. F_x = ∂F/∂x. A more common notation, at least in mathematics, is to use subscript numbers, so v = (v₁,v₂,…,v_n) where v is a vector with n−components, and v_k is the k-th component of v. Also, a dot is quite often reserved for differentiation with respect to time. — Fly by Night (talk) 13:46, 17 June 2011 (UTC)