An airplane's velocity with no wind is 580km/h with a bearing of N 60 Degrees.
The wind has a velocity 60km/h and is blowing S 45 degrees E. What is the true speed and bearing of the plane?
N 60 E (I assume) at 580
N component = 580 cos 60 =290 km/h
E component = 580 sin 60 = 502 km/h
now I assume you mean it is blowing TOWARD S 34 E
( in navigation we usually state wind direction FROM but we also say "heading" not "bearing" unless we are taking a bearing so I assume a non-navigator wrote the question)
wind drift S = 60 *.707 = 42.4 km/h
wind drift E = 60 *.707 = 42.4 km/h
total speed N = 290 - 42.4 = 248
total speed E = 503 + 42.4 = 545
speed = sqrt(248^2 + 545^2)
tan angle East of North = 545 / 248
Well, this calls for some mathematical clowning around! Let's calculate the true speed and bearing of the plane.
First, let's break down the wind's velocity into horizontal and vertical components. The wind is blowing S 45 degrees E, so the horizontal component would be 60 km/h multiplied by the cosine of 45 degrees, and the vertical component would be 60 km/h multiplied by the sine of 45 degrees.
Horizontal component = 60 km/h * cos(45 degrees) = approximately 42.43 km/h
Vertical component = 60 km/h * sin(45 degrees) = approximately 42.43 km/h
Now, let's add the horizontal components of the wind and the plane's velocity. Since the wind is blowing from the south to the north, we subtract the horizontal component of the wind from the plane's velocity.
Horizontal component of the true speed = 580 km/h - 42.43 km/h = approximately 537.57 km/h
Next, let's add the vertical components of the wind and the plane's velocity. Since the wind is blowing from the east to the west, we subtract the vertical component of the wind from the plane's velocity.
Vertical component of the true speed = 0 km/h - 42.43 km/h = -42.43 km/h
Now that we have the horizontal and vertical components of the true speed, we can use basic trigonometry to find the true speed and bearing.
True speed = square root of [(horizontal component)^2 + (vertical component)^2]
True bearing = arctan(vertical component / horizontal component)
True speed = square root of [(537.57 km/h)^2 + (-42.43 km/h)^2] = approximately 538.12 km/h
True bearing = arctan(-42.43 km/h / 537.57 km/h) = approximately -4.50 degrees
So, the true speed of the plane is approximately 538.12 km/h, and the true bearing is around -4.50 degrees. Just remember that these calculations should be taken with a pinch of clowning!
To find the true speed and bearing of the plane, we need to find the resultant velocity vector of the plane and the wind.
Step 1: Resolve the wind velocity into its north and east components:
The wind is blowing in a direction S 45 degrees E, which is the opposite direction of N 45 degrees W.
The north component of the wind velocity can be found using the sine function:
North component of wind velocity = 60 km/h * sin(45 degrees) = 42.43 km/h (rounded to two decimal places)
The east component of the wind velocity can be found using the cosine function:
East component of wind velocity = 60 km/h * cos(45 degrees) = 42.43 km/h (rounded to two decimal places)
Therefore, the wind velocity can be resolved into 42.43 km/h north and 42.43 km/h east.
Step 2: Add the north and east components of the wind velocity to the north and east components of the plane velocity to get the resultant velocity:
North component of resultant velocity = Plane's north component velocity + Wind's north component velocity
North component of resultant velocity = 580 km/h * cos(60 degrees) + 42.43 km/h = 290 km/h + 42.43 km/h = 332.43 km/h (rounded to two decimal places)
East component of resultant velocity = Plane's east component velocity + Wind's east component velocity
East component of resultant velocity = 580 km/h * sin(60 degrees) + 42.43 km/h = 502.92 km/h + 42.43 km/h = 545.35 km/h (rounded to two decimal places)
Therefore, the resultant velocity is 332.43 km/h north and 545.35 km/h east.
Step 3: Use the Pythagorean theorem to find the true speed of the plane:
True speed = √(resultant north component velocity^2 + resultant east component velocity^2)
True speed = √(332.43 km/h)^2 + (545.35 km/h)^2)
True speed = √(110,558.39 km^2/h^2 + 297,238.42 km^2/h^2)
True speed = √407,796.81 km^2/h^2
True speed = 639.22 km/h (rounded to two decimal places)
Step 4: Find the bearing of the plane using trigonometry:
Bearing = arctan(resultant east component velocity / resultant north component velocity)
Bearing = arctan(545.35 km/h / 332.43 km/h)
Bearing = arctan(1.6408)
Using a calculator, the arctan(1.6408) ≈ 57.34 degrees
Therefore, the true speed of the plane is approximately 639.22 km/h, and the bearing is approximately N 57.34 degrees.
To find the true speed and bearing of the plane, we need to first find the actual velocity of the plane by taking into account the effect of the wind.
Let's break down the given information:
1. The airplane's velocity with no wind is 580 km/h with a bearing of N 60 degrees.
2. The wind has a velocity of 60 km/h and is blowing S 45 degrees E.
To calculate the true speed and bearing of the plane, we need to consider the components of the wind and the plane's velocity separately.
Step 1: Resolving the wind's components:
Given that the wind is blowing in the direction of S 45 degrees E, we can break it down into its north-south (S) and east-west (E) components.
The south component (S) can be calculated using trigonometry:
S = wind velocity * sin(wind direction)
S = 60 km/h * sin(45 degrees)
S ≈ 60 km/h * 0.7071
S ≈ 42.42 km/h (rounded to two decimal places)
The east component (E) can be calculated using trigonometry:
E = wind velocity * cos(wind direction)
E = 60 km/h * cos(45 degrees)
E ≈ 60 km/h * 0.7071
E ≈ 42.42 km/h (rounded to two decimal places)
So, the wind's components are S = 42.42 km/h (south component) and E = 42.42 km/h (east component).
Step 2: Finding the resultant velocity of the plane:
To find the resultant velocity of the plane, we need to add the components of the plane's velocity (with no wind) and the wind's components.
The north component (N) of the plane's velocity remains unchanged at 580 km/h.
The east component (E) of the plane's velocity can be calculated using trigonometry:
E = wind's east component + plane's east component
E = 42.42 km/h + 0 km/h (since the plane's velocity has no component in the east direction)
E ≈ 42.42 km/h (rounded to two decimal places)
The resultant velocity of the plane (R) can be calculated using the Pythagorean theorem:
R = sqrt(N^2 + E^2)
R = sqrt(580 km/h^2 + 42.42 km/h^2)
R ≈ sqrt(336400 km^2/h^2 + 1796.9764 km^2/h^2)
R ≈ sqrt(338196.9764 km^2/h^2)
R ≈ 581.27 km/h (rounded to two decimal places)
So, the true speed of the plane is approximately 581.27 km/h.
Step 3: Finding the true bearing of the plane:
To find the true bearing of the plane, we need to determine the direction of the resultant velocity.
The angle (θ) can be calculated using trigonometry:
θ = arctan(N/E)
θ = arctan(580 km/h/42.42 km/h)
θ ≈ arctan(13.68)
θ ≈ 86.45 degrees (rounded to two decimal places)
Since the wind is blowing from the south and the plane's bearing is N 60 degrees, we need to subtract the angle θ from the plane's bearing to find the true bearing.
True bearing = Plane's bearing - θ
True bearing = 60 degrees - 86.45 degrees
True bearing ≈ -26.45 degrees (rounded to two decimal places)
So, the true bearing of the plane is approximately -26.45 degrees. Note that negative angles indicate a direction in the opposite sense (i.e., counter-clockwise) from the reference direction (N).
Therefore, the true speed of the plane is approximately 581.27 km/h with a bearing of -26.45 degrees.