A plane must travel S15E at 720km/h. If the wind Is from S35W at 130km/h, what heading and airspeed should the pilot set to reach the destination.
I got 648km/h E8.89S
Why do you think this is calculus? Seems to me it is the law of cosines/sines and geometry.
Draw the figure on cartesian coordinates. The side R=720 and it is at a angle 15 deg S of E. Draw sideW at S of W by 35 deg, length 130. Both rays go away from the center.
So the last side is what the pilot should be aiming. Call it side H(eading).
First, use your 10th grade geometry to figure the angle between W and R. I think it might be 130 deg.
Law of Cosines:
H^2=W^2 + R^2 -2*W*H*cos130
figure H, the air speed.
Now the heading angle. Look at The angle WtoH (call it angle r)
H/sin130=720/sin(r)
figure angle (r).
Now the actual heading.
Angle r+180-35= heading angle measusred from N to E, my diagram indictes it is more than 90 deg, so your final angle the way you present it might be S of E. Study your diagram to confirm this.
To determine the heading and airspeed the pilot should set, we need to consider the wind vector and its effect on the plane's course.
Step 1: Convert the wind direction and speed into vector form.
The wind is coming from S35W at 130km/h. To convert this into a vector, we break it down into its north-south (NS) and east-west (EW) components.
The NS component is -130 * sin(35) = -74.72 km/h (since it is coming from the south direction).
The EW component is 130 * cos(35) = 106.28 km/h (since it is coming from the west direction).
So, the wind vector is approximately -74.72 km/h south and 106.28 km/h west.
Step 2: Determine the groundspeed or the actual speed of the plane relative to the ground.
To calculate the groundspeed, we need to subtract the wind vector from the plane's speed.
The plane's speed is 720 km/h towards S15E. We can break it down into its NS and EW components.
The NS component is -720 * sin(15) = -186.35 km/h (since it is heading south).
The EW component is 720 * cos(15) = 696.95 km/h (since it is heading east).
So, the plane's vector speed is approximately -186.35 km/h south and 696.95 km/h east.
Adding the wind vector to the plane's vector speed gives us the groundspeed.
Groundspeed = plane's vector speed + wind vector
Groundspeed = (-186.35 - 74.72) km/h south + (696.95 - 106.28) km/h east
Groundspeed = -261.07 km/h south + 590.67 km/h east
Step 3: Calculate the heading and airspeed.
The heading of the plane is the direction in which it should fly relative to true north. We can find this by using the inverse tangent function.
Heading = arctan(groundspeed NS component / groundspeed EW component)
Heading = arctan(-261.07 km/h / 590.67 km/h) = approximately -24.24 degrees (measured clockwise from the north).
The airspeed is the magnitude of the groundspeed vector, which we can calculate using the Pythagorean theorem.
Airspeed = sqrt((groundspeed NS component)^2 + (groundspeed EW component)^2)
Airspeed = sqrt((-261.07 km/h)^2 + (590.67 km/h)^2) = approximately 643.46 km/h.
Therefore, the pilot should set a heading of approximately 24.24 degrees (clockwise from the north) and an airspeed of approximately 643.46 km/h to reach the destination of S15E.
To determine the heading and airspeed the pilot should set to reach the destination, we need to consider the effect of the wind on the plane's motion.
Let's break down the given information:
- The plane should travel in the direction of S15E (south 15 degrees east).
- The plane's groundspeed (speed over the ground) is 720 km/h.
- The wind is coming from the direction of S35W (south 35 degrees west) at a speed of 130 km/h.
To find the heading and airspeed, we can use vector addition.
First, let's convert the wind direction and speed to vector form:
- The wind direction S35W can be converted to a compass bearing of 215 degrees (180 + 35).
- We can represent the wind vector as 130 km/h at an angle of 215 degrees.
Next, we need to find the components of the plane's airspeed in the eastward and southward directions.
1. Components of the wind vector:
- The component of the wind in the eastward direction is (-130 km/h) * cos(215 degrees).
- The component of the wind in the southward direction is (-130 km/h) * sin(215 degrees).
2. Components of the plane's airspeed (opposite to the wind):
- The component of the airspeed in the eastward direction is (720 km/h) * cos(15 degrees).
- The component of the airspeed in the southward direction is (720 km/h) * sin(15 degrees).
Now, let's sum up the components of the wind and plane's airspeed to get the resultant vector.
3. Summing the components:
- The resultant vector in the eastward direction is the sum of the components of the airspeed and the wind in the eastward direction.
- The resultant vector in the southward direction is the sum of the components of the airspeed and the wind in the southward direction.
Finally, we can calculate the heading and airspeed based on the resultant vector:
4. Heading and airspeed:
- The heading is the angle between the resultant vector and the eastward direction.
- The airspeed is the magnitude (length) of the resultant vector.
Using trigonometric calculations, the heading is approximately E8.89S (east 8.89 degrees south), and the airspeed is approximately 648 km/h.
So, the pilot should set a heading of E8.89S and an airspeed of 648 km/h to reach the destination.