A plane flies at 150m/s at 35 degrees east of north. It encounters a wind from the west blowing 26 m/s. What is the true course of the plane? (speed and direction)
north speed = Vn = 150 cos 35
east speed = Ve = 150 sin 35 + 26
speed = sqrt (Vn^2+Ve^2)
heading = tan-1(Ve/Vn)
To find the true course of the plane, we need to combine the plane's velocity with the wind's velocity vector. Let's break it down step by step.
Step 1: Assess the components of the plane's velocity:
The plane's velocity can be decomposed into two components: the northward component and the eastward component. We can calculate these by using trigonometry, specifically the sine and cosine functions.
Given:
Magnitude of plane's velocity (speed): 150 m/s
Angle east of north: 35 degrees
The northward component of the plane's velocity is calculated as follows:
Northward component = Speed × sine(angle)
Northward component = 150 m/s × sine(35 degrees)
The eastward component of the plane's velocity is calculated as follows:
Eastward component = Speed × cosine(angle)
Eastward component = 150 m/s × cosine(35 degrees)
Step 2: Calculate the wind component:
The wind is blowing from the west (opposite direction of east), so its eastward component will be negative.
Given:
Magnitude of wind's velocity (speed): 26 m/s
Since the wind is blowing from the west, we know that the wind's direction is 270 degrees (west in relation to north).
The eastward component of the wind's velocity is calculated as follows:
Eastward component of wind = -Speed × cosine(wind's angle)
Eastward component of wind = -26 m/s × cosine(270 degrees)
Step 3: Combine the components:
Now, we need to add the corresponding components together to get the resultant vectors.
The northward component of the total velocity:
Northward component = Northward component of plane's velocity + Northward component of wind's velocity
The eastward component of the total velocity:
Eastward component = Eastward component of plane's velocity + Eastward component of wind's velocity
Step 4: Calculate the true course:
The true course is the direction of the total velocity vector. We can determine the angle by using the inverse tangent function.
True course angle = inverse tangent (Northward component / Eastward component)
Finally, to get the speed and direction, we use the Pythagorean theorem:
Speed = square root((Northward component)^2 + (Eastward component)^2)
By following these steps and plugging in the given values, you will find the true course (speed and direction) of the plane.