A plane has an airspeed of 520 mph. The pilot wishes to fly on...

Question

A plane has an airspeed of 520 mph. The pilot wishes to fly on a bearing of 310°. A wind of 37 mph is blowing from a bearing of 212°. In what direction should the pilot fly, and what will be her ground speed?

Accepted Answer

Hey, everyone here we are given a pilot flying a jet at an air speed of 400 kilometers per hour. Wants to fly. Maintaining a bearing of 307 degrees, determine the direction in which he should fly. If the wind is blowing at 500 kilometers per hour on a bearing of 220 degrees. Also find out the ground speed. We have four answer choice options, answer choices A through D where the bearing is either 330 degrees or 300 degrees. And the ground speed is either 410 kilometers per hour or 401 kilometers per hour. To begin this problem, we need to start by creating a sketch containing our given information. So to start our sketch, we need to draw two coordinate systems which are diagonal to each other and the top axis is our north axis. So just drawing our two coordinate systems focusing first on our bottom coordinate system, we first want to label its origin as point A. Now we are first given that the jet is flying at a speed of 400 kilometers per hour. So we can extend an arrow from our point A to the origin of our second coordinate system. And we can label this second origin as point C. And now we want to label this arrow as its given value of 400 kilometers per hour. Now, we want to maintain a bearing of 307 degrees. So we need to go 307 degrees starting at our north axes on our bottom coordinate system. So we go 307 degrees clockwise. And at this point, we need to draw a second arrow which extends at this angle from point A to the first quadrant of our second coordinate system. And we will label the head of this arrow at this point as point B. And since this arrow is again 307 degrees from our north axes, it tells us that the angle between this arrow and the north axes, the small angle here is then 53 degrees. And since this angle is 53 degrees, that tells us that the angle between our 400 kilometer per hour arrow and the south axes of our second coordinate system, that angle is also going to be 53 degrees. And our next piece of information is that the wind is blowing at 50 kilometers per hour on a bearing of 220 degrees. So now we are focusing on our second coordinate system and we need to travel 220 degrees clockwise from the north axes. And doing so we see we end up in the third quadrant. And so we just want to first label this angle as 220 degrees. And now in the third quadrant, we want to extend a dash line to the origin. And now extending this line into the first quadrant, we have our arrow which connects again from the origin to point B and this arrow represents our 50 kilometers per hour wind speed. And with that, we have sketched our given information. And the last thing we can identify on our second coordinate system is that the angle in between the 50 kilometer per hour arrow and the north axis is 40 degrees. And so with all of the information we have gathered, we can simplify our sketch by just creating a single triangle. So we can start with our point A of the triangle and extend an arrow upward which will point to our point B which tells us this arrow is our unknown ground speed. And now going back to point A, we extend another arrow outwards and this will be our 400 kilometers per hour speed. And lastly, to finish our triangle, we first want to label the other point here as point C. And now extending an arrow from point C to point B, we have our 50 kilometer per hour speed, wind speed. And lastly from our sketch, we can find that our angle C for the triangle is 87 degrees. Now, with the sketch of our simplified triangle, we can determine our ground speed by utilizing the law of cosines to find our length A B. So from the law of cosines, we know that length A B, which is our ground speed is equal to the square root of length AC squared plus length BC squared minus two multiplied by length ac multiplied by length BC multiplied by cosine of angle C. And now substituting in the values from our triangle, we have that A B is equal to the square root of 400 squared plus 50 squared minus two multiplied by 400 multiplied by 50 multiplied by cosine of 87 degrees. And now just evaluating, we find that A B, the length of A B or the ground speed is approximately equal to 400.5079 kilometers per hour. And now just rounding to the nearest whole number, we find that our ground speed is approximately equal to 401 kilometers per hour. And now that we have found the ground speed, we can use the law of signs to work towards finding the direction the pilot should fly. So utilizing the law of signs, we have that length BC divided by sine of angle A is equal to length A B divided by sine of angle C. Now substituting in the values we already have, we have 50 divided by sine of angle A is equal to 400.5079 divided by sine of 87 degrees. Now solving first for sine of A, we have sine of A is equal to 50 multiplied by sine of 87 degrees divided by 400.5079. Now taking sine inverse of both sides, we find that angle A is equal to sine inverse of 50 multiplied by sine of 87 degrees divided by 400.5079. Evaluating we find that angle A is approximately equal to 7.1617 degrees, which is approximately equal to seven degrees. Now, in order to find the direction which the pilot should fly, we need to take the value of angle A and subtract it from the bearing value that the pilot wants to maintain. So we just have 307 degrees minus seven degrees which simplifies to just 300 degrees. And with this, we have found the direction at which he should fly as well as the ground speed. And with these values, we are left here with answer choice C where again the bearing is 300 degrees and the ground speed is 401 kilometers per hour. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

A plane has an airspeed of 520 mph. The pilot wishes to fly on a bearing of 310°. A wind of 37 mph is blowing from a bearing of 212°. In what direction should the pilot fly, and what will be her ground speed?

Key Concepts

Vector Addition of Velocities

Bearing and Angle Conversion

Trigonometric Resolution of Vectors

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