An airplane flies with a speed of 425 mph and a heading of 63°. If the heading of the wind is 24°and the speed of the wind is 31 mph, what is the heading of the plane and the ground speed?
I've done a couple questions like this but still a little fuzzy on setting it up, any help would be greatly appreciated.
V = 425mi/h @ 63o + 31mi/h @ 24o
X = 425*cos63 + 31*cos24 = 221.3 mi/h.
Y = 425*sin63 + 31*sin24 = 391.3 mi/hr.
tanA = Y/X = 391.3/221.3 = 1.7681
A = 60.5o. = Direction of plane.
V = X/cosA = 221.3/cos60.5 = 431 mi/h =
Velocity of plane.
To solve this problem, we can use vector addition.
First, let's break down the velocity of the airplane and the velocity of the wind into their components. The speed of the airplane can be broken down into two components: the horizontal component (or ground speed) and the vertical component (or vertical speed/climbing or descending speed). The direction of the heading represents the angle between the airplane's velocity vector and the horizontal axis.
Given:
Speed of the airplane (also called ground speed, or horizontal component): 425 mph
Heading of the airplane: 63°
To find the ground speed, we need to find the horizontal component of the velocity. We can use trigonometry to determine this. The horizontal component can be calculated using the equation:
Ground speed = Speed of the airplane * cos(Heading of the airplane)
So, plugging in the values:
Ground speed = 425 mph * cos(63°)
Ground speed ≈ 425 mph * 0.448
Ground speed ≈ 190.4 mph (rounded to one decimal place)
Now let's consider the wind. The speed of the wind is 31 mph, and its heading is 24°. We can find the horizontal component of the wind speed by using trigonometry again:
Horizontal component of wind speed = Speed of the wind * cos(Heading of the wind)
Plugging in the values:
Horizontal component of wind speed = 31 mph * cos(24°)
Horizontal component of wind speed ≈ 31 mph * 0.912
Horizontal component of wind speed ≈ 28.3 mph (rounded to one decimal place)
Now, to find the heading of the plane after accounting for the wind, we need to add the heading of the airplane and the heading of the wind.
Heading of the plane after accounting for the wind = Heading of the airplane + Heading of the wind
Plugging in the values:
Heading of the plane after accounting for the wind = 63° + 24°
Heading of the plane after accounting for the wind ≈ 87° (rounded to one decimal place)
Thus, the heading of the plane after accounting for the wind is approximately 87°, and the ground speed (horizontal component) of the plane is approximately 190.4 mph.