Find the bearing and airspeed for a plane to fly 630 miles due north in 3 hours if the wind is blowing from a direction of 318* and is blowing at 15mph. Approximate the bearing to the nearest degree and the speed to the nearest mile per hour.
Vpw = 630mi/3h = 210m1/h, Due North
Vpw = Vp + Vw = 210i
Vp + 15[318o] = 210i
Vp + 15*Cos318+15*sin318 = 210i
Vp + 11.15 -10i = 210i
Vp = -11.15 + 210i + 10i = -11.15 + 220i
,Q2.
Tan A = 220/-11.15 = -19.73094
A = -87.1o = 87.1o N. of W. = 92.9o CCW
from + x-axis. = 2.9o W. of N. = Direction.
Vp = 220/sin92.9 = 220.3 mi/h
To find the bearing and airspeed for the plane to fly 630 miles due north in 3 hours, we can use vector addition.
First, we need to find the effective wind vector. We can do this by subtracting the wind vector (direction and speed) from the desired direction of travel (due north) and magnitude (630 miles in 3 hours).
1. Find the components of the wind vector:
- Direction: 318 degrees
- Speed: 15 mph
To find the horizontal component of the wind vector, we can multiply the wind speed by the cosine of the wind direction:
Horizontal Component = 15 mph * cos(318 degrees)
To find the vertical component of the wind vector, we can multiply the wind speed by the sine of the wind direction:
Vertical Component = 15 mph * sin(318 degrees)
2. Find the effective northward component of the plane's velocity:
Northward Component = 630 miles / 3 hours = 210 mph
To find the horizontal component of the plane's velocity, we can subtract the horizontal component of the wind vector from the northward component:
Horizontal Component = 210 mph - Horizontal Component (from step 1)
To find the vertical component of the plane's velocity, we can subtract the vertical component of the wind vector from 0 (since the plane is flying due north):
Vertical Component = 0 - Vertical Component (from step 1)
3. Find the magnitude and direction of the resulting velocity:
Magnitude = sqrt((Horizontal Component)^2 + (Vertical Component)^2)
Direction = arctan(Vertical Component / Horizontal Component)
4. Round the magnitude to the nearest mile per hour and the direction to the nearest degree to obtain the final accuracy requested.
Calculating the values step by step:
Horizontal Component = 15 mph * cos(318 degrees) ≈ -12.58 mph
Vertical Component = 15 mph * sin(318 degrees) ≈ -6.92 mph
Horizontal Component = 210 mph - (-12.58 mph) ≈ 222.58 mph
Vertical Component = 0 - (-6.92 mph) ≈ 6.92 mph
Magnitude = sqrt((222.58 mph)^2 + (6.92 mph)^2) ≈ 222.82 mph
Direction = arctan(6.92 mph / 222.58 mph) ≈ 1.77 degrees
Therefore, the approximate bearing is 2 degrees and the approximate airspeed is 223 mph.
To find the required bearing and airspeed of the plane, we need to consider the effect of the wind.
Step 1: Determine the groundspeed of the plane
The groundspeed of the plane is the resulting speed of the plane when factoring in the wind. We can calculate it using the Pythagorean theorem.
Groundspeed = √(airspeed^2 + wind speed^2)
Given the wind speed is 15 mph and the airspeed is unknown, we'll use a variable (let's call it 'x') for the airspeed.
Groundspeed = √(x^2 + 15^2)
Step 2: Calculate the course angle
The course angle refers to the direction or bearing at which the plane should fly relative to true north.
Using trigonometry, we can find the course angle using the following formula:
Course angle = arccosine(wind speed / groundspeed)
Since we already know the wind speed (15 mph) and the groundspeed (which involves the airspeed), we can substitute them into the formula.
Course angle = arccosine(15 / Groundspeed)
Step 3: Substitute the given values and solve for the airspeed
We know that the plane travelled 630 miles in 3 hours, which means the groundspeed is the distance divided by the time.
Groundspeed = 630 miles / 3 hours
Groundspeed = 210 miles per hour
Now, with the groundspeed determined, we can substitute it back into the equation for the groundspeed:
210 = √(x^2 + 15^2)
Solving this equation will give us the value for 'x,' which represents the airspeed of the plane.
Step 4: Calculate the bearing and approximate the values
Once we have the airspeed, we can substitute it back into the equation for the course angle to find the bearing:
Course angle = arccosine(15 / Groundspeed)
Bearing = 360 - Course angle
Finally, we round the bearing to the nearest degree and the airspeed to the nearest mile per hour.
By following these steps, you can determine the approximate bearing and airspeed for the plane to fly 630 miles due north in 3 hours with the given wind conditions.