an airplane is flying a compass of 340 degrees at 325 mph. A wind is blowing with the bearing 300 degrees at 30 mph. Find the component form of the velocity, and find the actual ground speed and direction of the plane.
V = 325mph[340o] + 30mph[300o]
X = 325*cos340 + 30*cos300 = 320.4 mph =
Hor. component.
Y = 325*sin340 + 30*sin300 = -137.1 mph
= Ver. component.
TanA = Y/X = -137.1/320.4 = -42802
A = -23.17o,CW = 336.8 CCW.
V = 320.4/cos336.8 = 348.6mph[336.8o]
V= <325 cos110, 325 sin110>
W=<40cos130, 40sin130>
Resultant =V+W
Actual ground speed = magnitude of resultant = √(-136.87)^2+(336.04)^2 = 362.85 mph
To find the component form of the velocity, we need to break down the velocity of the airplane into its horizontal and vertical components.
Given information:
- Airplane's compass bearing: 340 degrees
- Airplane's speed: 325 mph
- Wind's bearing: 300 degrees
- Wind's speed: 30 mph
First, let's find the horizontal and vertical components of the airplane's velocity.
Horizontal Component:
The horizontal component is given by multiplying the airplane's speed by the cosine of its compass bearing.
Horizontal Component = Airplane speed × cosine(Compass Bearing)
Horizontal Component = 325 mph × cos(340 degrees)
Vertical Component:
The vertical component is given by multiplying the airplane's speed by the sine of its compass bearing.
Vertical Component = Airplane speed × sine(Compass Bearing)
Vertical Component = 325 mph × sin(340 degrees)
Next, let's find the horizontal and vertical components of the wind's velocity.
Horizontal Component:
The horizontal component is given by multiplying the wind's speed by the cosine of its bearing.
Horizontal Component = Wind speed × cosine(Wind Bearing)
Horizontal Component = 30 mph × cos(300 degrees)
Vertical Component:
The vertical component is given by multiplying the wind's speed by the sine of its bearing.
Vertical Component = Wind speed × sine(Wind Bearing)
Vertical Component = 30 mph × sin(300 degrees)
Now, let's add the horizontal components and vertical components of both velocities to find the total horizontal and vertical components.
Total Horizontal Component = Airplane's Horizontal Component + Wind's Horizontal Component
Total Vertical Component = Airplane's Vertical Component + Wind's Vertical Component
Finally, the actual ground speed of the plane is given by the magnitude of the resultant vector (total velocity components), and the direction is given by the angle it makes with the positive horizontal axis.
Ground Speed = √(Total Horizontal Component^2 + Total Vertical Component^2)
Direction = arctan(Total Vertical Component / Total Horizontal Component)
Plug in the values and calculate to find the answers.
To find the component form of the velocity, we need to determine the horizontal and vertical components separately.
Step 1: Finding the horizontal component of velocity:
The airplane is flying at a compass bearing of 340 degrees, which means the angle between its direction and the positive x-axis is 340 degrees. To find the horizontal component of the velocity, we can use the cosine function:
Horizontal component (Vx) = Velocity (325 mph) * cos(Bearing angle - Wind angle)
Vx = 325 mph * cos(340 degrees - 300 degrees)
Vx = 325 mph * cos(40 degrees)
Vx ≈ 248.63 mph
Step 2: Finding the vertical component of velocity:
To find the vertical component of the velocity, we can use the sine function:
Vertical component (Vy) = Velocity (325 mph) * sin(Bearing angle - Wind angle)
Vy = 325 mph * sin(340 degrees - 300 degrees)
Vy = 325 mph * sin(40 degrees)
Vy ≈ 220.07 mph
Therefore, the component form of the velocity is (248.63 mph, 220.07 mph).
To find the actual ground speed of the plane, we need to combine the effects of the wind with the airplane's velocity. We can use vector addition to do this.
Ground speed = √(Vx^2 + Vy^2)
Ground speed = √((248.63 mph)^2 + (220.07 mph)^2)
Ground speed ≈ √(61,860.37 mph^2 + 48,442.33 mph^2)
Ground speed ≈ √(110,302.7 mph^2)
Ground speed ≈ 331.96 mph
To find the direction of the plane's velocity or the ground track, we can use the inverse tangent function:
Ground track angle = tan^(-1)(Vy / Vx)
Ground track angle = tan^(-1)(220.07 mph / 248.63 mph)
Ground track angle ≈ tan^(-1)(0.8843)
Ground track angle ≈ 41.59 degrees
Therefore, the actual ground speed of the plane is approximately 331.96 mph, and the ground track angle is approximately 41.59 degrees.