An airplane is flying on a compass heading (bearing) of 340 degrees at 325 mph. A wind is blowing with the bearing 320 degrees at 40 mph. Find the component form of the velocity of the airplane. then find the actual ground speed and direction of the plane
Airplane velocity A=(325 cos110, 325 sin110)
Wind velocity W=(40 cos130, 40 sin 130)
resultant = A+W = <-136.87, 336.04>
Actual ground speed = √(-136.87)^2 +(336.04)^2 = 362.85 mph
angle theta= arc tan (-336.04/136.87) = 112.16
bearing = 337.84
Troll
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To find the component form of the velocity of the airplane, we need to find the horizontal and vertical components separately.
Step 1: Calculate the horizontal component of the velocity.
We can use the cosine function to find the horizontal component. The formula is:
Horizontal component = Velocity * Cosine(Bearing)
Horizontal component = 325 mph * Cos(340 degrees)
Horizontal component = 325 mph * Cos(-20 degrees) [Since Cos(340 degrees) = Cos(-20 degrees)]
Now, we need to convert degrees to radians since most programming languages use radians for trigonometric functions. We can use the formula:
radians = degrees * (pi/180)
degrees = -20
radians = -20 * (pi/180)
radians = -20 * (pi/180) = -0.349 radians
So, the horizontal component is:
Horizontal component = 325 mph * Cos(-0.349 radians)
Horizontal component = 325 mph * (0.9415)
Horizontal component ≈ 306.19 mph
Step 2: Calculate the vertical component of the velocity.
Similar to the horizontal component, we can use the sine function to find the vertical component. The formula is:
Vertical component = Velocity * Sine(Bearing)
Vertical component = 325 mph * Sin(340 degrees)
Vertical component = 325 mph * Sin(-20 degrees) [Since Sin(340 degrees) = Sin(-20 degrees)]
Using the same conversion from degrees to radians as above:
degrees = -20
radians = -20 * (pi/180)
radians = -20 * (pi/180) = -0.349 radians
So, the vertical component is:
Vertical component = 325 mph * Sin(-0.349 radians)
Vertical component = 325 mph * (-0.3379)
Vertical component ≈ -109.67 mph
Therefore, the component form of the velocity of the airplane is:
(306.19 mph, -109.67 mph)
To find the actual ground speed and direction of the plane, we can add the wind vector to the velocity vector of the airplane.
Step 3: Calculate the horizontal component of the wind.
We can use the same formula as before:
Horizontal component of the wind = Wind speed * Cosine(Bearing)
Horizontal component of the wind = 40 mph * Cos(320 degrees)
Converting degrees to radians:
degrees = 320
radians = 320 * (pi/180)
radians = 320 * (pi/180) = 5.585 radians
So, the horizontal component of the wind is:
Horizontal component of the wind = 40 mph * Cos(5.585 radians)
Horizontal component of the wind = 40 mph * (0.9638)
Horizontal component of the wind ≈ 38.55 mph (rounded to two decimal places)
Step 4: Calculate the vertical component of the wind.
Using the same formula as before:
Vertical component of the wind = Wind speed * Sine(Bearing)
Vertical component of the wind = 40 mph * Sin(320 degrees)
Converting degrees to radians:
degrees = 320
radians = 320 * (pi/180)
radians = 320 * (pi/180) = 5.585 radians
So, the vertical component of the wind is:
Vertical component of the wind = 40 mph * Sin(5.585 radians)
Vertical component of the wind = 40 mph * (0.2667)
Vertical component of the wind ≈ 10.67 mph (rounded to two decimal places)
Now, we can add the components of the velocity of the airplane and the wind together to get the actual ground speed and direction of the plane.
Actual ground speed = magnitude of (velocity of the airplane + wind)
Actual ground speed = sqrt((horizontal component of velocity + horizontal component of the wind)^2 + (vertical component of velocity + vertical component of the wind)^2)
Actual ground speed = sqrt((306.19 mph + 38.55 mph)^2 + (-109.67 mph + 10.67 mph)^2)
Actual ground speed = sqrt((344.74 mph)^2 + (-99 mph)^2)
Actual ground speed = sqrt(118739.5076 + 9801)
Actual ground speed ≈ sqrt(128540.5076)
Actual ground speed ≈ 358.66 mph (rounded to two decimal places)
To find the direction, we can use the tangent function:
Direction = atan2((vertical component of velocity + vertical component of the wind), (horizontal component of velocity + horizontal component of the wind))
Direction = atan2(-109.67 mph + 10.67 mph, 306.19 mph + 38.55 mph)
Direction = atan2(-99 mph, 344.74 mph)
Direction ≈ -16.59 degrees (rounded to two decimal places)
Therefore, the actual ground speed of the plane is approximately 358.66 mph, and the direction is approximately -16.59 degrees.
Vh = hor = 325cos340 + 40cos320,
Vh = 305 +30.6 = 335.6mph.
Vv = ver = 325sin340 + 40sin320,
Vv = -111.2 -25.7 = -136.9mph.
tanA = Vv/Vh = -136.9 / 335.6 = 0.4080,
A = - 22.2deg CW = 337.8 deg. CCW.
R = Vh / cosA = 335.6 / cos337.8 = 362.5mph @ 337.6 deg.