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

1. Jan 12, 2013

### 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?

2. Jan 12, 2013

### Staff: Mentor

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

3. Jan 12, 2013

### 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.

Last edited: Jan 12, 2013
4. Jan 12, 2013

### PHYSMajor

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.

5. Jan 12, 2013

### PHYSMajor

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.

Last edited: Jan 12, 2013
6. Jan 12, 2013

### Staff: Mentor

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.

7. Jan 12, 2013

### Staff: Mentor

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

8. Jan 12, 2013

### mycotheology

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.

9. Jan 12, 2013

### PHYSMajor

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.

10. Jan 12, 2013

### Staff: Mentor

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?
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.

11. Jan 12, 2013

### PHYSMajor

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.

12. Jan 12, 2013

### cepheid

Staff Emeritus
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.

13. Jan 12, 2013

### Staff: Mentor

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.

14. Jan 12, 2013

### 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

Cheers,
Bobbywhy

15. Jan 12, 2013

### Staff: Mentor

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.

16. Jan 13, 2013

### Staff: Mentor

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.

17. Jan 13, 2013

### Staff: Mentor

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.

18. Jan 13, 2013

### Staff: Mentor

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.

19. Jan 13, 2013

### PHYSMajor

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.

20. Jan 13, 2013

### cepheid

Staff Emeritus
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.