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2.25 ভেক্টর relative velocity math problems | আপেক্ষিক বেগের বৃষ্টি ছাতার সমস্যা- part -03| – Transcript
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- The speaker explains umbrella-holding math problems in rain involving passenger velocity and rain velocity.
- When rain falls vertically at 5 meters per second with no wind, a stationary passenger holds the umbrella vertically.
- Adding wind velocity from east to west alongside vertical rain and pedestrian movement introduces relative velocity calculations and resultant vectors.
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In the last video, we did a few math problems related to relative velocity. In today's video, we will look at a few more problems on relative velocity. If I categorize the relative velocity problems that you find difficult or troublesome into two parts, the first category would be umbrella-holding problems in rain, and the second category would be car windshield wetting problems in rain. Believe me, these problems are very easy. But many students find these problems quite
difficult. One reason for this difficulty is similar to trying to learn how to form sentences in English without first learning the alphabet; it's going to feel very difficult. Similarly, if you try to jump into complex math without mastering the small, basic problems, this simple topic will seem very hard to you.
Another reason is the lack of "feel" for the subject. Meaning, if you are only doing it on paper and cannot feel the situation realistically using common sense, it will seem difficult. So, the first step to making these math problems easy is to bring in that feeling. That is, we need to examine whether the math we are doing actually holds true in real life, and another task will be to gradually move from small, basic problems toward the difficult ones. So, if you watch the
math problems I solve in this video in order and with full attention, I hope you won't have any more trouble with relative velocity problems. Okay, let's begin. So, first, let's look at the umbrella-holding problems in the rain. I have already solved one umbrella-holding problem for you before . So, now we will solve many such problems. Previously, we saw a boy just running in the rain. Meaning, there was the passenger's velocity and the rain's velocity. Now, we will look at cases where there is passenger velocity, rain
velocity, and also wind velocity. So, let's start with a very simple math problem. Our first problem states that rain is falling vertically at 5 meters per second. In the first case, it is mentioned that there is no wind. No wind. Meaning, we will not include wind in our calculations here. Our first problem asks how a stationary passenger should hold an umbrella to stay
protected from the rain. The second problem asks how the passenger should hold the umbrella if they start walking at 10 meters per second. The third one asks if the angle of the umbrella with the vertical increases as the passenger's velocity increases. Explain this mathematically. So let's get started. Our first task will be to draw the diagram. Rain is falling vertically
at 5 meters per second. So, this is the rain. We have drawn this rain. Represented by Vr. The velocity is 5 meters per second. Now look, this is the flat ground. A passenger is standing here. The passenger is standing like this. So the passenger is standing still. The passenger has no velocity. How is the rain falling?
Vertically. The question asks how a passenger standing still should hold an umbrella. Is there any need to do math for this? You are standing still. Rain is falling vertically onto your head. So, if you have an umbrella, to protect yourself from the rain, you will hold the umbrella vertically right over your head. There is no need for any math at all. So why did I give this? There is a reason why I gave it. We will understand in a little while. So if we write the answer in this case, what will we write? Hold the umbrella
vertically. Number one is done. Hold the umbrella vertically. In this case, we don't even need to do any calculation. Now let's look at the second case. What does the second one say? How should the umbrella be held if the passenger starts walking at 10 meters per second? First, I told you to get a feel for it and think about the real-world scenario. You were standing still and you held the umbrella
vertically over your head. Now the rain is falling. You have started walking in this direction. So, as soon as you start walking or running forward like this. Notice that in real life, we tilt the umbrella at an angle with the vertical. That is, the umbrella is tilted in the direction we are running.
So let's see what happens. I told you that the velocity of the rain is the relative velocity of the rain with respect to the passenger. That is, when the passenger moves forward, the velocity of the rain relative to the passenger will seem to be coming from this direction. So, for relative velocity, we said we have to reverse the velocity of the reference point.
Since it is the velocity of the rain relative to the passenger, we will reverse the passenger's velocity. So notice that the passenger was moving this way with a velocity of 10 meters per second, this is the passenger's velocity. Since the rain's velocity is relative to the passenger, we reverse the passenger's velocity here, considering the raindrop. Meaning, we will not keep it this way anymore. So, the passenger's velocity in this direction will be what? 10 meters per second. How is the rain falling? The rain is falling vertically. At what velocity? At a velocity of 5 meters per
second. So the first task was to reverse, and the second task was to find the resultant after reversing. So look, the reversal task is done. We have done such math before. So this velocity vector is no longer on this side. After reversing, the resultant of these two will come along this line.
The resultant will come along this line . So the rain will hit the passenger here from this direction. So if the passenger wants to protect themselves from the rain by holding an umbrella, they must hold the umbrella with the vertical. Look, with the vertical line, they have to hold the umbrella tilted at an angle theta. This means to protect themselves from the rain, they will now hold the umbrella. We have been asked to find the value of this angle theta for holding the umbrella.
Look, how will they hold the umbrella? So, I was saying the angle to hold the umbrella with the vertical is theta. Now, let's find theta in a very short way. Can we write down tan theta? Let's look at tan theta carefully. This angle is theta. This is 10, and what was this ? This was 5. So, if we shift this up.
If I bring it to this side, this will also be 5. So, tan theta equals perpendicular by base, which is 10 by 5. Therefore, theta equals tan inverse 2, and calculating this gives 63.43. Degrees; meaning, to stay protected from the rain, one must hold the umbrella at a 63.43-degree angle with the vertical. I hope we understood this concept. Now look at what number three says: analyze mathematically if the
angle of the umbrella with the vertical increases when the pedestrian increases their speed. It would be great if I had an umbrella; camera, do we have the umbrella? Let's see, look, this is it. When rain falls vertically, I hold the umbrella straight above my head like this. This was the first case, vertically; now, if I start walking this way, notice, I am moving in this direction. So when I move this way, I tilt the umbrella like this. At what angle with the vertical do I tilt it? I
tilt it at this 63.43-degree angle. Think about it for a moment. If I increase my speed in the direction I am walking—that is, if I go faster than before—wouldn't I tilt the umbrella even more like this? So, you see, the umbrella has to be tilted at a larger angle with the vertical line. That is what the problem asks: if the pedestrian increases speed, does the umbrella angle with the vertical increase? Explain and analyze this mathematically. So, let's see if we can
explain and analyze it mathematically. Suppose for the third case, imagine our speed was 10, and now we make the speed 15. So if the speed is 15, then after reversing it, the velocity on this side will be 15. It will be on this side. And the rain falling is at a speed of 5 meters per second. Then, the resultant of these two will strike the pedestrian like this. So, if the pedestrian wants to be protected from the rain, they must now hold the umbrella with the vertical. The umbrella must be held at
an angle of theta. This angle is that theta angle. So, what was our target? To find this theta angle. Look, if this is 5, then this is 5. So if we calculate this angle, let's write it at once: theta equals tan inverse perpendicular 15, divided by base 5, that is, tan inverse 3. Calculate it, and it will come to 71.56 degrees. So, now to be protected from the rain, one must tilt the umbrella at a larger angle of 71.56 degrees with the vertical. Earlier, when the speed was
10, at what angle did the umbrella need to be held? At a 63.43-degree angle. So , you see, the more he increases his speed, the more he has to tilt the umbrella at a greater angle from the vertical to stay protected from the rain. I hope you have understood this concept. Try to feel this thing in a practical sense. Then you will be able to understand it even more easily. Now, we are going to do the same type of math. But notice, earlier there was no wind. So, what are we going to do now?
We will add wind to it and then see how he needs to hold the umbrella. The second one is vertical rain at 5 meters per second and wind blowing from east to west at 2 meters per second. Notice earlier there was no wind. Now I have added wind. And in which direction is the wind blowing? It is blowing from the east to the west. With what velocity? With a velocity of 2 meters per second. And the rain is falling the way it was falling before. Vertical
rain at 5 meters per second. Now, the first math problem asks how a stationary traveler should hold the umbrella. So, let's see how a stationary traveler should hold the umbrella. Let's draw it. First, let's draw the rain. The rain is falling vertically with what velocity? With a velocity of 5 meters per second. And what else is there? Wind. In which direction? From east to west. Let's say this direction is the west direction.
We call this direction the east direction. So, the wind is blowing from east to west at 2 meters per second. Now, in which direction should a stationary observer hold the umbrella to stay protected from the rain? That is our question. Think about it for a moment first. Imagine the rain is falling vertically. There is no wind.
In that case, how would a stationary observer hold the umbrella? Since the rain is falling vertically, he would hold the umbrella vertically over his head. Which is what we did in the previous math problem. Now the question is, if there is vertical rain and a wind comes from this side, can the traveler still hold the umbrella vertically? No, he cannot. Because the wind will hit the raindrops that are
falling vertically. Then the raindrops will no longer be able to fall along that vertical path. That is, the path of the rain will deflect, and it will fall along the resultant velocity of the wind and the rain. So, notice that if a traveler is standing still, he will no longer hold the umbrella vertically over his head. He will then definitely hold the umbrella at a theta angle from the vertical by tilting it.
Think about it yourself; imagine you are standing still in one place. The rain is falling vertically. You are holding the umbrella vertically. But if suddenly you see a gust of wind coming from that direction. Will the rain still fall on your head vertically then ? Then, if you have an umbrella in your hand, you will certainly tilt it in the direction from which the gust of wind is coming. Are you getting the feel of it? So we saw that for a stationary observer, if the rain falls vertically, they will hold the umbrella vertically
over their head. But if there is wind, you must tilt the umbrella in the direction from which the wind is blowing. So, this thing can be felt, right? It can be understood and explained, can't it? Okay. Now we need to calculate. At what angle should the umbrella be held? For a stationary observer, these raindrops will fall with a resultant velocity of the rain's velocity and the wind's velocity. Then this passenger must hold the umbrella along this direction. So, at what angle with the vertical? At an angle of theta
(θ). Our target is to find this theta angle. So similarly, if this is five, then along this direction it is five and this one is two. So if we calculate the value of the theta angle, it is tan inverse of 2/5 with the vertical. Then the value of this angle will be 21.80 degrees. Meaning, they must hold the umbrella at this angle with the vertical. Note that in this case, we did not reverse anyone's velocity. We did not calculate the relative velocity of one with respect to another. Where we first reverse the velocity of the one with respect to whom the velocity
is being calculated. And then find the resultant after reversing it. We did not do this task this time. Now, there is something for you to think about here. In previous math problems, when it was the velocity of rain with respect to a boy or a passenger. We did reverse the passenger's velocity. But in this case, when dealing with the velocity of wind and rain, notice we did not reverse any velocity. We directly calculated the resultant of the wind velocity and rain velocity, and the rain will fall with this resultant velocity of wind and rain. So from this, one thing can be said: if it