Relative Velocity in Two Dimensions: Rain and Wind

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle from the monsoon. There is no wind, and the rain is falling straight down. You start walking fast with your umbrella, and suddenly, the rain seems to come at you from the front! But it is falling straight down, right? So where is it coming from? And where should you tilt your umbrella: forward, backward, or straight up? Today we settle it with one simple subtraction. And these rain and umbrella questions come up often in JEE papers. Let's begin.

0:32Rain, as the walking man sees it

So, look at the figure. On the right, our man walks forward. The orange arrow below him is his velocity, v m vector: the small arrow on top means a vector, and the small m stands for man. On the left, the blue arrow pointing straight down is the rain's velocity, v r vector, the small r for rain. Rain straight down, man straight ahead, okay? Now think back to the last lesson. The man is moving, so he does not see the rain's real velocity. So tell me, how does the rain move, as the man sees it? Yes, exactly! The relative velocity. Just like in one dimension, we subtract. v r m vector equals v r vector minus v m vector. The small r slash m means rain relative to man. In the triangle, minus v m vector is the orange arrow pointing backwards, opposite to his walk. Add it to the blue rain arrow, and you get the green arrow, v r m vector: the rain as he feels it. Clear? Now look at the green rain arrows on the right. They slant down and come at him from the front! That is the rain in our puzzle. So the umbrella must point against v r m vector, into the rain he sees, not against the rain as the ground sees it. Got it?

1:45Vertical rain: how much to tilt

So now, how much to tilt? In the triangle, theta is the angle at the top, between the blue vertical arrow and the green one. Opposite to theta is the orange side, the man's speed. Next to it is the blue side, the rain's speed. So, what is tan theta? Opposite by adjacent, correct! tan theta equals v m by v r; the line means divided by. No arrows now: v m and v r are just the speeds, the lengths of the arrows. And remember, this theta is measured from the vertical, okay? Write this down. Now a classic trap. The rain falls straight down. So you hold the umbrella straight up, right? No, no, careful! Straight down is the rain as the ground sees it. You need the rain as you see it, the green arrow, and it comes from the front. So tilt the umbrella forward, into the rain. Puzzle solved! And a topper's habit: never guess the tilt. Draw this triangle every time, and the answer is right in front of you.

2:43Same idea in the air: aircraft and wind

Alright, the same idea works in the sky. For an aircraft, v ground vector equals v air vector plus v wind vector. v air vector is the plane's velocity through the air, what it would have in still air. Add the wind's velocity, and you get v ground vector, its real velocity over the ground. Now rearrange it: v air vector equals v ground vector minus v wind vector. See? It is again a relative velocity, the plane relative to the moving air. Same subtraction as the rain. Exactly the same. Now think like a pilot. The wind blows towards the east, and you want to travel due north. Where should you point the nose of the plane? West of north, yes! Partly into the wind. The westward part of the plane's velocity cancels the eastward wind. So over the ground, it travels due north. Simple as that.

3:39Example 1: rain at 4 m/s, man at 3 m/s, umbrella angle

Right, on to Example one. Rain falls vertically at four metres per second; the m s to the power minus one means metres per second. The man walks at three metres per second. How should he hold his umbrella? Pause the video, and try it first. Done? Okay, let's check. First, how fast does the rain hit him? Four plus three, seven metres per second, right? No, no, careful! These two are at right angles: rain down, and minus v m vector backwards, like the triangle in the figure. So use Pythagoras. v r m is the square root of four squared plus three squared; the tick-shaped sign is the square root. Root of sixteen plus nine: five metres per second. No arrow on v here, so it is just the speed. Now the angle. tan theta equals v m by v r, that is three by four. So theta is tan inverse three by four; the small minus one means inverse. That is about thirty seven degrees: the wavy equals sign means approximately, and the little circle means degrees. So, the rain seems to come at him from the front, exactly our puzzle! He holds the umbrella tilted forward, at thirty seven degrees to the vertical.

4:57Example 2: actual rain velocity from two walking speeds

Okay? Now a harder one, JEE level. Walking at three kilometres per hour, that is km h to the power minus one, a man finds the rain falling vertically. Walking at six, he finds it falling at forty five degrees. Find the actual velocity of the rain. Pause and try. Hint: give the rain two unknown parts. Done? Okay, let's check. Let the rain's velocity, v r vector, be a comma minus b, in brackets. Here a is the part along his walk, and b is the part downwards; down is negative, so minus b. With me? Now, at three kilometres per hour, the rain looks vertical to him. So the actual rain is falling vertically, right? No, no! That is the JEE trap. It only looks vertical to him. What he sees is the relative velocity: a minus three, comma minus b. Vertical means no sideways part, so a minus three is zero; the double arrow means so. So a is three, and the real rain has a sideways part along his walk! Now he walks at six. The relative velocity is three minus six, comma minus b. That is minus three, comma minus b. And he sees it falling at forty five degrees. So, how do the two parts of this relative velocity compare? Equal in size, yes! At forty five degrees to the vertical, the sideways part and the downward part are the same. So b equals three. So both parts of the real rain are three. Its speed, no arrow now, is the square root of three squared plus three squared, that is three root two kilometres per hour. Clear? And the direction? The parts are equal, so it falls at forty five degrees to the vertical, slanting towards his direction of walking. See, the rain was never vertical at all! It only looked vertical. Remember, what you see is always the relative velocity. If you solved this one yourself, well done, that was a proper JEE question!

7:03Quick revision: rain seen by the man, umbrella, aircraft

So, quick revision. Rain as you see it: v r m vector equals v r vector minus v m vector, and the umbrella points against it. For vertical rain, tan theta equals v m by v r, from the vertical, tilted forward. And for an aircraft, v ground vector equals v air vector plus v wind vector. Now go and try the practice problems on this page. You will find them easy. All the best!