Why does vertically falling rain make a slanted steaks on a window?

[PHYSMajor]

I recently solved a question related to the problem below, but am having trouble getting an intuition for the problem.

Suppose an automobile is traveling at a constant horizontal velocity, u, and it's raining. There is no wind, so the raindrops do not have an initial horizontal velocity, just a vertical one. However, when the rain reaches the window, it is "given" a horizontal velocity. This horizontal velocity, as measured relative to a point on the ground, is the same as that of the automobile's. However, if we take our reference frame to be a point on the automobile, then the horizontal velocity of the rain should be zero, no? Thus the rain would only have a vertical velocity, and should not appear to be slanted from the perspective of someone sitting inside the car.

So essentially, my question is: Why does vertically falling rain make slanted streaks on the side of a window?

 

[Doc Al]

You have it backwards. Relative to the ground, the rain has no horizontal component of velocity. Relative to the car it does.

 

[mycotheology]

Its all a matter of air resistance isn't it. The water droplets cannot oppose the air resistance as well as the car can. If you stick a piece of string out the window, the string won't hang vertically, it will hang slightly horizontally even though its horizontal velocity with respect to the car is 0. Thats only because it can't oppose the air resistance as well as the car. If you hang a piece of paper with the same mass as the string, it won't hang as all, it will be completely horizontal because its affected by air resistance to a much greater extent than the string.

Also in the case of water droplets, a major factor to be considered is the intermolecular forces between the water and the window. If it was raining hexane, I bet the streaks would be much more horizontal because there would be much weaker intermolecular forces between it and the glass. Hanging a piece of string out the window is a better example because you don't have to consider intermolecular forces attaching the second object to the car.

 

[PHYSMajor]

You have it backwards. Relative to the ground, the rain has no horizontal component of velocity. Relative to the car it does.

Initially, the rain would not have a horizontal component. Once it is "on the car," the horizontal velocity of the rain relative to the rain would be the same as that of the car relative to the ground. This would mean that the horizontal velocity of the rain relative to the car is zero. I know I am doing something wrong, but I don't understand what.

 

[PHYSMajor]

Its all a matter of air resistance isn't it. The water droplets cannot oppose the air resistance as well as the car can. If you stick a piece of string out the window, the string won't hang vertically, it will hang slightly horizontally even though its horizontal velocity with respect to the car is 0. Thats only because it can't oppose the air resistance as well as the car. If you hang a piece of paper with the same mass as the string, it won't hang as all, it will be completely horizontal because its affected by air resistance to a much greater extent than the string.

The problem assumes that there is no air resistance. Apologies for not mentioning that.

If the car was moving at a constant velocity, and a vertical piece of string, say, hangs from the roof of the car, the string would stay vertical. It would not be slanted. I just can't bring myself to understand why the rain wouldn't behave the same way if seen from someone inside the car.

 

[Doc Al]

Initially, the rain would not have a horizontal component.

With respect to the ground the velocity of the rain has no horizontal component. Not just 'initially', but at all times.

Once it is "on the car," the horizontal velocity of the rain relative to the rain would be the same as that of the car relative to the ground.

The car is moving at some horizontal velocity with respect to the ground; the rain is not. With respect to the car, the rain does have a horizontal velocity. If the car is moving at 60 mph east with respect to the road, then the rain has a horizontal component of 60 mph west with respect to the car.

 

[Doc Al]

If the car was moving at a constant velocity, and a vertical piece of string, say, hangs from the roof of the car, the string would stay vertical. It would not be slanted. I just can't bring myself to understand why the rain wouldn't behave the same way if seen from someone inside the car.

The rain clouds are not traveling along inside the car, are they?

 

[mycotheology]

The problem assumes that there is no air resistance. Apologies for not mentioning that.

If the car was moving at a constant velocity, and a vertical piece of string, say, hangs from the roof of the car, the string would stay vertical. It would not be slanted. I just can't bring myself to understand why the rain wouldn't behave the same way if seem from someone inside the car.

Ah right, then in that case I think intermolecular forces would be the main factor behind non vertical streaking. The water droplets don't instantaneously merge with the glass, they can only attach to the glass with adhesive forces determined by the intermolecular forces between H2O and SiO2. The car is accelerating (or it at least had to accelerate to reach its current velocity) but the water droplet isn't. If it was raining superglue, then the droplets would be attached to the window so strongly (after they solidified at least) that they can be considered as the same object as the car, and thus accelerate with the car but in the case of water droplets, they're not truly part of the car, they are only clinging on with dipole-dipole interactions.

 

[PHYSMajor]

With respect to the ground the velocity of the rain has no horizontal component. Not just 'initially', but at all times.

The car is moving at some horizontal velocity with respect to the ground; the rain is not. With respect to the car, the rain does have a horizontal velocity. If the car is moving at 60 mph east with respect to the road, then the rain has a horizontal component of 60 mph west with respect to the car.

Once the rain is on the window, it is moving at the same speed as the car. If the car is moving at 60mph, then the rain is also moving at 60mph. The 60mph of the car was measured with respect to the ground, so it should be the same for the rain. So how can it be that the horizontal velocity of the rain wrt the ground is zero at all times?

Secondly, I mentioned that the rain is on the window, and is traveling at the same horizontal velocity as the car. Then how can the rain have a 60mph with respect to the car? This is like saying that if you are running along a track at x mph, and a dog is running beside you at x mph (these velocities are measured with respect to the ground), the dog's velocity is x mph with respect to you. We know that this isn't true because the dog is running right beside you. If it's velocity with respect to you was x mph, the dog would not be running beside you, but ahead of you.

For some reason, I am beginning to get the feeling that that my understanding of relative motion is wrong. I don't know where I am wrong though.

 

[Doc Al]

Once the rain is on the window, it is moving at the same speed as the car. If the car is moving at 60mph, then the rain is also moving at 60mph. The 60mph of the car was measured with respect to the ground, so it should be the same for the rain. So how can it be that the horizontal velocity of the rain wrt the ground is zero at all times?

Ah, so you are talking about after the rain has hit the car and come to rest? Do you realize that before the rain actually strikes the car that it is moving at an angle with respect to the car, thus of course it streaks at an angle?

Secondly, I mentioned that the rain is on the window, and is traveling at the same horizontal velocity as the car.

Only after the drops come to rest on the surface of the car, if they every do. But the interesting thing is the speed and angle at which they hit the car.

 

[PHYSMajor]

Ah right, then in that case I think intermolecular forces would be the main factor behind non vertical streaking. The water droplets don't instantaneously merge with the glass, they can only attach to the glass with adhesive forces determined by the intermolecular forces between H2O and SiO2. The car is accelerating (or it at least had to accelerate to reach its current velocity) but the water droplet isn't. If it was raining superglue, then the droplets would be attached to the window so strongly (after they solidified at least) that they can be considered as the same object as the car, and thus accelerate with the car but in the case of water droplets, they're not truly part of the car, they are only clinging on with dipole-dipole interactions.

Wouldn't this mean that the drops are not traveling at the same horizontal velocity as the car? The problem I was doing in the textbook said that we are to assume they are traveling at the same horizontal velocity as the car. I think this is more a question of relative motion than of intermolecular interactions. What you say makes perfect sense though. It's just that the question assumes that intermolecular interactions is not the case. In a more realistic situation, it would be.

 

[cepheid]

Are you talking about raindrops streaking across a side window?

You seem be assuming that as soon as the raindrop comes in contact with the car, it instantaneously gains a horizontal component equal to the car's and therefore should begin to move forward with it. Not necessarily...

Edit: Doc Al basically said this two posts up. Sorry.

 

[Doc Al]

The problem I was doing in the textbook said that we are to assume they are traveling at the same horizontal velocity as the car.

Are you sure it said velocity and not speed?

If the car is moving at 60 mph with respect to the road, then the rain will have a horizontal velocity of 60 mph with respect to the car.

Can you please give the name of your textbook and the problem number.

 

[Bobbywhy]

PHYSMajor,

The answer to your question, “Why does vertically falling rain make slanted streaks on the side of a window?” is simple: The water drop adheres to the window and gravity pulls it downward. The wind rushing by drags it towards the rear of the car. The resultant of the two forces is a slanted path downwards and backwards.

Cohesion: Water is attracted to water
Adhesion: Water is attracted to other substances
http://ga.water.usgs.gov/edu/adhesion.html

Cheers,
Bobbywhy

 

[Chestermiller]

There is something missing from the analyses that have appeared so far. I am a fluid mechanics guy, so I think I can clear up the issue. When a drop first hits the car, not all parts of the drop take on the car velocity instantaneously. Only the part of the drop at the very interface with the car body takes on the car velocity. This is the so-called "no slip" boundary condition of fluid mechanics. Other parts of the drop, because of their inertia, are still traveling with the velocity they had before hitting the car. Because of viscous stresses, the zero velocity effect at the surface penetrates into the drop, and eventually the entire body of water that originally comprised the drop achieves the velocity of the car. In practice, I do think that air drag also plays an important role in retarding the rate at which the water achieves the car velocity.

 

[Doc Al]

The answer to your question, “Why does vertically falling rain make slanted streaks on the side of a window?” is simple: The water drop adheres to the window and gravity pulls it downward. The wind rushing by drags it towards the rear of the car. The resultant of the two forces is a slanted path downwards and backwards.

I'd say it was even simpler. "Vertically" falling rain makes slanted streaks on the side windows because with respect to the car the rain isn't falling vertically. No need to involve wind here. The car is moving; that's all you need.

I don't think we need to get into the fluid dynamics or adhesion effects of the drop/glass interaction to explain this simple effect! As PHYSMajor suspects, the problem is one of understanding reference frames.

 

[Chestermiller]

I'd say it was even simpler. "Vertically" falling rain makes slanted streaks on the side windows because with respect to the car the rain isn't falling vertically. No need to involve wind here. The car is moving; that's all you need.

I don't think we need to get into the fluid dynamics or adhesion effects of the drop/glass interaction to explain this simple effect! As PHYSMajor suspects, the problem is one of understanding reference frames.

If it weren't for the "no slip boundary condition" and surface tension, the drop would not leave a streak on the window. It would simply slide in its entirety along the window, without leaving a residue. As for air drag (wind), I guess you've never been in a car where a kid throws up out the front passenger window, and the vomit splashes all over the rear passenger window. Drops are much smaller than globs of vomit and are affected much more strongly by air drag.

 

[Doc Al]

If it weren't for the "no slip boundary condition" and surface tension, the drop would not leave a streak on the window. It would simply slide in its entirety along the window, without leaving a residue.

The key point is that it slides at an angle due to the relative velocity.

As for air drag (wind), I guess you've never been in a car where a kid throws up out the front passenger window, and the vomit splashes all over the rear passenger window. Drops are much smaller than globs of vomit and are affected much more strongly by air drag.

The problem ignores air resistance. Or did you miss that?

Don't complicate a simple problem.

 

[PHYSMajor]

Are you sure it said velocity and not speed?

If the car is moving at 60 mph with respect to the road, then the rain will have a horizontal velocity of 60 mph with respect to the car.

Can you please give the name of your textbook and the problem number.

Okay, so I have two texts here, with similar questions. I shall quote one of the texts, and try to show you which parts confuse me, and why. I'm almost certain I haven't understood reference frames properly.

Problem: "A car travels due east with a speed of 50.0km/h. Rain-drops are falling at a constant speed vertically with respect to the Earth. The traces of the rain on the side windows of the car make an angle of 60.0° with the vertical. Find the velocity of the rain with respect to (a) the car and (b) the Earth."

Solution: (a) - I drew a rt. angle triangle with a 60.0° angle between the vertical and the horizontal. I'm supposed to find the length of the hypotenuse, which would give me the velocity of the raindrops. This gives sin (60.0°) = (50.0km/h)/hyp, so hyp. = (50.0km/h)/sin(60.0) = 57.7km/h.

(b) - The book says that the answer is the length of the vertical, which is simply (50.0k/h)/tan (60.0°) = 28.9km/h.

Now, I don't understand why the velocity of the raindrops with respect to the Earth is the vertical, and not the hypotenuse of my triangle. Likewise, why isn't the velocity of the drops with respect to the car the vertical instead of the hypotenuse? The answers seem reversed to me. Going back to the analogy of the dog and person running side by side at a velocity (a m/s) with respect to the earth, the velocity of the dog with respect to the person would be 0, as the origin is placed on the person and the person itself is moving. With this analogy, wouldn't the drop, which has a horizontal velocity because it is on the car, have a horizontal velocity with respect to the earth, meaning that with respect to the earth, the drop has both a horizontal and a vertical velocity, and thus the velocity in part (b) should be the length of the hypotenuse and not the vertical? I get the impression that once the drop is on the car we are to assume that it has the same horizontal velocity as the car, and because the horizontal velocity of the car was measured with respect to the earth, this should also be the case with the drop.

The way I've phrased the question may make it difficult to understand. It's the best I could do.

 

[cepheid]

The velocity of the drop relative to the Earth is the drop velocity seen by an observer who is stationary with respect to the earth. So it's the velocity seen by an observer standing on the side of the road,for example. This is vertical, since it's explicitly stated in the problem that the drops are falling straight down (no wind).

The velocity relative to the car is the velocity seen by an observer who is stationary relative to the car. Eg the driver. I'll use the notation v_a/b to mean "velocity of 'a' relative to 'b'". In this case, to get the drop velocity in the car frame, we just add velocities like so

v_drop/car = v_drop/earth + v_earth/car

In words: the velocity of the drop relative to the car is equal to the velocity of the drop relative to the Earth plus the velocity of the Earth relative to the car. Note that this is a vector sum.

In his case, since v_drop/earth is vertically
downward, and v_earth/car is westward, the resultant has both downward and westward components. The drop trajectory appears slanted from the car.

 

[Chestermiller]

The key point is that it slides at an angle due to the relative velocity.

The problem ignores air resistance. Or did you miss that?

Don't complicate a simple problem.

No, I didn't miss it. I just think it's silly to ignore an effect that in real life would be so important. My main issue would be with the problem statement.

 

[PHYSMajor]

The velocity of the drop relative to the Earth is the drop velocity seen by an observer who is stationary with respect to the earth. So it's the velocity seen by an observer standing on the side of the road,for example. This is vertical, since it's explicitly stated in the problem that the drops are falling straight down (no wind).

It's true that the v_drop/earth (using your notation here) is vertical before the drops reach the side window. However, when they are on the side window and have "acquired" the horizontal velocity of the car while the car is moving, doesn't v_drop/earth now have a horizontal component as well?

The velocity relative to the car is the velocity seen by an observer who is stationary relative to the car. Eg the driver. I'll use the notation v_a/b to mean "velocity of 'a' relative to 'b'". In this case, to get the drop velocity in the car frame, we just add velocities like so

v_drop/car = v_drop/earth + v_earth/car

Wouldn't the correct formula be v_drop/earth = v_car/earth + v_drop/car? This is the formula I have been using all along. The reason is because I am assuming the Earth is stationary, and the car is moving relative to the earth. The drop is also moving relative to the earth, but after it is on the car's window, it would have the horizontal velocity of the car. This would mean (as it seems to me) that relative to the earth, the velocity of the drop is the vector sum of the velocity of the car relative to the earth, and the drop relative to the earth.

Please take note that I am not concerned with the velocity of the drop before it has reached the window. Before it is "on" the window, it only has a vertical velocity, I understand that part perfectly. However, after it is on the window, the velocity of the drop should have a horizontal component, in addition to its original vertical component, no?

 

[Doc Al]

Please take note that I am not concerned with the velocity of the drop before it has reached the window.

But you should be.

Before it is "on" the window, it only has a vertical velocity, I understand that part perfectly.

Vertical with respect to the ground, not the car. You want to know with what velocity the drop hit the car with respect to the car.

However, after it is on the window, the velocity of the drop should have a horizontal component, in addition to its original vertical component, no?

You are confusing yourself by worrying about the fate of the drops once they are on the car. Of course, once the drops are on the car and moving along with it (which will be affected by wind and such), they will now have a horizontal velocity with respect to the road since the car has accelerated them. I suggest that you forget about that and first understand the velocity of the drops as seen from the car just before they hit.

 

[cepheid]

It's true that the v_drop/earth (using your notation here) is vertical before the drops reach the side window. However, when they are on the side window and have "acquired" the horizontal velocity of the car while the car is moving, doesn't v_drop/earth now have a horizontal component as well?

For the second time: this doesn't happen instantaneously. The drops are going to streak across the window at an angle because they hit it at an angle, because that is their velocity relative to the car while falling. This is the point Doc Al has been trying to make for two pages now.

Wouldn't the correct formula be v_drop/earth = v_car/earth + v_drop/car? This is the formula I have been using all along.

Your equation is not different from mine, since v_car/earth = -v_earth/car.

Please take note that I am not concerned with the velocity of the drop before it has reached the window. Before it is "on" the window, it only has a vertical velocity, I understand that part perfectly. However, after it is on the window, the velocity of the drop should have a horizontal component, in addition to its original vertical component, no?

Eventually the drop velocity it will gain a horizontal component (in the Earth frame) but not right away. Hence, in the Earth frame, the car's surface slides horizontally past the drop for a while, before friction gets rid of their relative horizontal motion, and the drop starts being carried along with the car.

EDIT: In the car frame, the drop strikes at an angle and streaks along the car body at that angle, but it's continuously slowing while doing so. Eventually it comes to a stop (no more horizontal motion across the car body) and just continues falling straight down the car body (unless friction is enough to kill this vertical component too).

 

[A.T.]

If it weren't for the "no slip boundary condition" and surface tension, the drop would not leave a streak on the window.

And let's not forget the big bang. Without that, no window, no water so no streak.

 

[jtbell]

The rain clouds are not traveling along inside the car, are they?

If the car is being driven by Joe Btfsplk, the rain clouds might well be traveling along above the car!

http://www.ojaipost.com/wp-content/uploads/2012/06/Joe-Btfsplk.jpg

http://en.wikipedia.org/wiki/Joe_Btfsplk

 

[sophiecentaur]

It might be better to start with a simpler model than a raindrop on a car. Say a falling ball bearing and how that might appear to the passengers, looking sideways - work up from there. People have been throwing in more and more factors and it just gets too confusing if you don't deal with one thing at a time.