Solve each problem. See Examples 5 and 6.

Question

Distance and Direction of a Motorboat A motorboat sets out in the direction N 80° 00′ E. The speed of the boat in still water is 20.0 mph. If the current is flowing directly south, and the actual direction of the motorboat is due east, find the speed of the current and the actual speed of the motorboat.

Accepted Answer

Everyone in this problem, we have a yacht sailing in the direction south 59 degrees east. If the speed of the current is zero, the yacht sails at a speed of 37 kilometers per hour. But in this case, we have the resultant direction of the yacht in the east because we have a current that is flowing directly north. They were asked to determine the actual speed of the yacht and the speed of the current. For given four answer choices in this problem options, A through D each containing a different speed for the current and for the uh we're gonna come back to these answer choices at the end of the problem. So let's start by drawing a little picture of what we have going on. So we have a yacht sailing south 59 degrees east. So we can imagine drawing our coordinates. We're gonna start at the center and we're sailing south 59 degrees east. So south is straight downwards and we're gonna go 59 degrees towards the east, which is towards the right, this is the direction we're sailing in. And again, the angle between the vertical and this direction is 59 degrees. Hm. Oh, we know that this direction that we're traveling in with no current, we have a speed of 37 kilometers per hour, but we actually have a current and the current is flowing directly north. So if we imagine adding that current, we can add it tip to tail with our direction of our yacht. Right? It's gonna have some speed VC. And we know that the resulting direction of the yacht is east, the resulting direction of the yacht is going to point straight to the east. Ok. And so the speed here, we're gonna call it VYC, the speed of the yacht in the current. And what we can see is that these three together form a nice right triangle and we have a yacht in the current traveling straight east, the current traveling straight north. So those two make a 90 degree angle with each other. Ok. So we wanna determine these two currents or speed, sorry, the speed of the current and the speed of the yacht in that current, we have a 90 degree triangle. So we can use our trigonometric ratios, we have the hypotenuse. So all we need is one angle and that angle we can find pretty easily, we can see that we have this angle of 59 degrees, ok. So the angle of 59 degrees plus the angle inside of our triangle have to form a 90 degree angle and they have to add up to 90. So that means that the angle inside of the triangle must be 31 degrees and 90 degrees minus 59 degrees gives us 31 degrees. So now we can go ahead and use our trigonometric ratios. And we are gonna start by finding the speed of the current VC. This is the opposite side to the angle that we have. And so we're gonna use sign to relate though. So recall that sin of theta can be written as the opposite side divided by the hypotonic. We get that sign of 31 degrees is equal to VC divided by 37 kilometers per hour. We can multiply both sides by 37 kilometers per hour. We're gonna get that. The speed of the current is about 19.05 64 kilometers per hour. Ok. So that is the speed of the current. Now, we need to go ahead and find the actual speed of the yacht. So the speed of the yacht in that current. So that's our VYC value and we're gonna relate that through cosine because it is the adjacent side. So we're called that cosine of theta is the adjacent side divided by the hypotenuse. That means that cosine of 31 degrees is equal to VYC divided by 37 kilometers per hour, multiplying both sides by 37 kilometers per hour. We get that VYC is approximately equal to 31.71 52 kilometers per hour. And what we can see is that the actual speed of this yacht is less than the speed of the yacht if there is no current. And that makes sense because the direction of the current is not pointing in the direction that that yacht is traveling. But when that current starts to move, it opposes the motion a little bit and it's gonna slow that yacht down. All right, let's look at our answer choices and compare what we found. We are gonna round to three significant digits. And when we do that, we can see that the correct answer is going to be option D the speed of the current is 19.1 kilometers per hour and the actual speed of the yacht is 31.7 kilometers per hour. Thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Vector Addition and Resultant Velocity

Resolving Vectors into Components

Trigonometric Direction Angles

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