A plane is flying due South (270º) at 474.8 km/h. A wind blows from West to East (0°) at 46 km/h.
Find the direction of the plane's travel. Report your answer in degrees from 0° as a reference.
46 km/h [Horizontal]
--->
\. |
.\.|
..\v -474.8 km/h [Vertical]
We're trying to find the angle in the top left corner of the triangle above. We already have the lengths of the two legs, so we'd do this using tan.
Tan(x) = Opposite/Adjacent
Tan(x) = (-474.8)/(46) = -10.3217
x = Tan-1(-10.3217)
x = -84.47 degrees
To find the direction of the plane's travel, we need to consider the effect of the wind on the plane's movement.
Since the plane is flying due south (270º), we need to add the wind's direction (0°) to the plane's direction in order to find the resulting direction.
270º + 0º = 270º
Therefore, the direction of the plane's travel is 270°.
To find the direction of the plane's travel, we need to consider the effect of the wind on the plane's path.
First, let's break down the velocity of the plane and the wind into their horizontal (East-West) and vertical (North-South) components.
The plane's initial velocity of 474.8 km/h due South can be broken down as follows:
- Horizontal component: 474.8 km/h * sin(270°) = 0 km/h (since sin(270°) = -1, and the plane is flying South)
- Vertical component: 474.8 km/h * cos(270°) = -474.8 km/h (since cos(270°) = 0, and the plane is flying South)
Similarly, the wind's velocity of 46 km/h from West to East can be broken down as follows:
- Horizontal component: 46 km/h * cos(0°) = 46 km/h (since cos(0°) = 1, and the wind is blowing from West to East)
- Vertical component: 46 km/h * sin(0°) = 0 km/h (since sin(0°) = 0, and the wind is not blowing vertically)
Now, let's add up the horizontal and vertical components of the plane's velocity and the wind's velocity separately.
- Horizontal component: 0 km/h + 46 km/h = 46 km/h (this represents the net horizontal velocity)
- Vertical component: -474.8 km/h + 0 km/h = -474.8 km/h (this represents the net vertical velocity)
Next, we need to find the resultant velocity (the combined effect of the plane's velocity and the wind's velocity). The resultant velocity can be found using the Pythagorean theorem:
Resultant velocity = sqrt((Horizontal component)^2 + (Vertical component)^2)
= sqrt((46 km/h)^2 + (-474.8 km/h)^2)
= sqrt(2116 km^2/h^2 + 225069.04 km^2/h^2)
= sqrt(227185.04 km^2/h^2)
≈ 476.8 km/h
To find the direction of the plane's travel, we can use trigonometry. The direction can be found using the equation:
θ = arctan(Vertical component / Horizontal component)
In this case, θ = arctan((-474.8 km/h) / (46 km/h))
= arctan(-10.331)
≈ -84.4°
Since the angle must be measured with respect to 0° as a reference, we can add 90° to the obtained angle.
Angle with respect to 0° = -84.4° + 90°
= 5.6°
Therefore, the direction of the plane's travel is approximately 5.6° east of north (with 0° as a reference).