While driving north at 25 m/s during a rainstorm you notice that the rain makes an angle of 38° with the vertical. While driving back home moments later at the same speed but in the opposite direction, you see that the rain is falling straight down. From these observations, determine the speed and angle of the raindrops relative to the ground.

Video transcript

Hey, everyone in this problem, a cyclist rides on a rainy day at a steady speed of seven km/h along the x axis in the positive direction. The cyclist remarks that the traces of the raindrops are tilted at an angle of 26° clockwise from the vertical direction. Let's start with this piece of the problem before we move on to the rest. We're just gonna draw a little diagram. So we have our cyclists and draw them as a dot They are moving seven km/h in the positive direction in the X-axis. We're gonna say that that's to the right of the traces of the raindrops are tilted. So we have the vertical axis shown in a dotted line And the raindrops are tilted 26° clockwise from the vertical direction. So we have an angle of 26°. Ok. So this is the initial situation. Now back to the problem due to the bad weather conditions, she decides to make a u-turn, she moves now at the same steady speed along the X axis in the negative direction. On her way back, she observes that traces of the raindrop are now aligned with the vertical. Yeah. So after some time, there's bad weather, she's gonna turn around. She's still gonna be traveling at seven km/h. But now she's pointed to the left we've chosen right to be positive. So she has a velocity of negative seven km/h and the traces of the raindrops are lined with the vertical. Now, we're asked to calculate the speed of the raindrops and their inclination with respect to a fixed frame on earth. All right. So we have our diagram we've taken right to be positive. And let's also take upwards to be positive. Let me redraw this. So we're taking upwards and to the right as positive directions. Now, we're asked to find the speed, the velocity of the raindrops with respect to the earth. So V R E, we do, we write a general equation for this. Now, the raindrops are falling downwards we've chosen up as their positive Y direction. So we know in the Y direction the speed is going to be negative. What about the X direction? Well, when the cyclist is moving to the left, the raindrops are falling vertically or appear to be falling straight vertically. It means that those raindrops are also have a negative X speed or velocity. OK. So we're gonna expect that the X and Y components are both negative and that they make some angle with the ground. OK. The raindrops are not gonna be falling perfectly straight up and down to the ground. They're gonna make some angle with that fixed frame on earth. And so we're gonna write the velocity of the raindrops with respect to the earth as negative V science data I hat minus V data J. And what we wanna do is we wanna find the speed of rain drop with respect to earth. So we need to find the value of V. No way we need to find the value of data. Now, what do we know? We know that the speed of the raindrops or the velocity of the raindrops with respect to the cyclist are gonna be equal to the velocity of the raindrops with respect to the earth plus the velocity of the earth with respect to the cyclist. Now, the velocity of the earth with respect to the cyclist, we don't know we know the velocity of the cyclist with respect to the earth. OK. So recall that we can write this as the negative. So we have the velocity of the raindrop with respect to the cyclist is going to be equal to the velocity of the raindrop with respect to the earth minus the velocity of the cyclist with respect to the earth. And so we negate it and then we can flip the order. Now the velocity of the ranger up with respect to the cyclist when you first look at this problem, OK. It's very enticing to go ahead and write that we're told the angle that, that raindrop makes with respect to the cyclist, but we don't know the speed that it's following. And so that can get a little bit messy to write that turn out. So let's stick with this for now and we're gonna work through it and you'll see why we don't need to do that. So let's start with the initial case. OK. So the initial case is when the cyclist is moving to the right, the velocity of the raindrop with respect to the cyclist is going to be equal to the velocity of the raindrop with respect to the earth, which we've said is going to be negative V sign of theta I hat minus V cos theta J hat minus the velocity of the cyclist with respect to the earth. And the CYC cyclist is moving in the positive X direction seven kilometers an hour. And so we have seven kilometers per hour I have if we rearrange so that we group the I hat components in the J hat components, we get the velocity of the Raindrop with respect to the cyclists. In this initial case is negative seven km/h minus B find data I had minus V ghost data J Alright. No, I've said we didn't know the speed the raindrops are falling, but we were told the angle that they make Right, the rain drops in this initial case, make a 26° angle with the vertical. So we have a relationship between the I hat and J hat components. OK? Let me draw out a triangle and you can see what I mean. So our I A component is the X direction component. Our Jha component is the Y direction component. OK? Both are negative. So the velocity of the raindrop with respect to the cyclist is the hypo nose. We know that the angle that this makes between the hypotenuse and the J hat component is 26 degrees, which tells us that the tangent of 26 degrees must be equal to the opposite side, which is the I hat component divided by the adjacent side, which is the J hat component. So we have the 10 of 26 degrees must equal negative seven kilometers per hour minus V sine theta divided by negative V cos the OK. So the I hack component from the velocity of the raindrop with respect to the cyclist divided by the J hack component. And that's how we can use that information about the angle. Now, there's negative, we can divide by a negative throughout this equation. We can write it as seven km/h plus v sine data divided by V multiplied by co theta. And we're gonna call this equation one and we're gonna come back to this and we have two unknowns in this equation. We have V and we have the, but we've only used the initial situation for now. So if we go and do the same thing for the actor situation. We're gonna have two equations with two unknowns and we'll be able to solve. OK. So after the runner or the cyclist, sorry has made their u-turn, we have that the velocity of the raindrop with respect to the cyclist is going to be equal to. Well, it's still equal to the velocity of the raindrop with respect to the earth, which we've defined as negative V multiplied by selling the in the I hat direction minus V multiplied by coast data in the J hat direction minus the velocity of the cyclist with respect to the earth. So we have minus now, the cyclist is traveling in the negative direction. And so they have a velocity of negative seven kilometers per hour, not Jha but I had if we rearrange so that we group the I hat and J hat components, we have seven kilometers per hour minus V multiplied by sine data in the I hack component minus V multiplied by data in the J hat direction. OK? Is the velocity of the range up with respect to the cyclist after the u-turn. Now, after the u-turn, we're told that the rain drops are aligned with the vertical. What that means is the angle is 0°. If the raindrops are aligned with the vertical, it means that their motion is completely in the J hat direction and they have no I hat component of their motion. This tells us that seven kilometers per hour minus V sine theta must be equal to zero. OK. That I hat component of the raindrops velocity with respect to the cyclist has to be zero because they're falling vertically. All right. Now, we have another equation with the same two unknowns V and sine theta. We can rearrange for one of them. Let's say we rearrange and sol for B, we can write that V is equal to seven kilometers per hour divided by Sine theta. And we're gonna call this equation two. And now we have two equations, two unknowns. We can use the substitution method to solve for the unknowns. So let's start by substituting equation two in into equation one, This will give us 10 of 26° is equal to seven km/h plus V which is now given by seven km/h divided by that the multiplied by sine of theta and all of this is divided by V Which is seven km/h divided by sign of the multiplied by cosine of theta. Now, this looks pretty messy, but it's going to simplify nicely. So if we simplify, we get that tin of 26° is equal to, We have seven km/h everywhere. OK? We can divide right now. I'm just gonna leave it. We have seven km/h Plus seven km/h. OK? We have sine data divided by sine data. So those will cancel I'm gonna leave the seven km an hour for now. So we don't get confused with too many steps at once. And in the denominator, we have coast data divided by sine data. Recall that Sine data divided by coast data is 10. And so this is gonna be 1/10. So in the denominator, we have seven km/h Multiplied by 1/10 data. And the numerator, we have 14 km/h. In the denominator, we have seven km/h. And so we get that this is equal to two km/h multiplied by 10 data. And so the, We Wanna isolate data, we divide by two And let me take the inverse 10, it's gonna be 10 inverse oh 10 of 26° divided by two. And again, that looks a little bit messy, but we can plug this into our calculator and we're gonna get an angle of 13.7, And that is that angle of inclination that we were looking for. So we found one piece of the puzzle. We have one piece left to find, we have data. Now, we can substitute that back into either equation one or two to find V. We're gonna choose equation two because it's a little bit simpler. And I'm just gonna go back up to the top. So we have some more room to work and we have the, the velocity V was equal to or the Speed V was equal to seven km/h divided by sine of theta. In this case, we found theta was 13. degrees. So we have 17 kilometers per hour divided by sine of 13.705 degrees. And we get that the speed V Is equal to .54, km/h. So now we have our speed and our inclination, we've solved the problem. Ok. So that was a long process. What we needed to do was just work out each component of our velocity and compare them with the information we were given about the angles of inclination for both when the cyclist was going to the right. And when the CYCR cyclist made that u-turn and traveled to the left, if we look at our answer choices, We can see they're rounded to three significant digits. So with three significant digits, we found that the speed of the raindrops were 29.5 km/h with an inclination of 13.7° with the vertical with respect to that fixed frame on Earth. So we have answer choice. D thanks everyone for watching. I hope this video helped see you in the next one.