To solve this problem, we need to use vector addition. The airplane's speed and the wind's speed are vectors because they have both magnitude (speed) and direction.

Step 1: Break down the vectors into their components.

The airplane's velocity vector is entirely in the east direction. So, its components are:
Vax = 600 mph (east component)
Vay = 0 mph (north component, which is zero because the plane is not going north or south)

The wind's velocity vector needs to be broken down into its east and north components. The angle given is measured from the south, towards the east. So, we can use sine for the east component and cosine for the north component (which will be negative because it's going south).

Wx = 85 mph * sin(59°) = 70.4 mph (east)
Wy = 85 mph * cos(59°) = -51.5 mph (south, which is negative north)

Step 2: Add the components of the vectors.

The resultant east component is the sum of the east components of the airplane and wind vectors:
Rx = Vax + Wx = 600 mph + 70.4 mph = 670.4 mph

The resultant north component is the sum of the north components of the airplane and wind vectors:
Ry = Vay + Wy = 0 mph - 51.5 mph = -51.5 mph

Step 3: Calculate the magnitude and direction of the resultant vector.

The magnitude (speed) is the square root of the sum of the squares of the components:
R = sqrt(Rx^2 + Ry^2) = sqrt((670.4 mph)^2 + (-51.5 mph)^2) = 672.3 mph

The direction is the arctangent of the north component divided by the east component. Remember to adjust the angle based on the quadrant. Since the east component is positive and the north component is negative, the angle is in the fourth quadrant:
θ = atan(Ry/Rx) = atan(-51.5 mph / 670.4 mph) = -4.4°

However, since the angle is measured from the east towards the north, we need to add 360° to the negative angle:
θ = -4.4° + 360° = 355.6°

So, the airplane's actual speed is 672.3 mph, and it's flying at an angle of 355.6° measured from the east, towards the north.

Question

To solve this problem, we need to use vector addition. The airplane's speed and the wind's speed are vectors because they have both magnitude (speed) and direction.

Step 1: Break down the vectors into their components.

The airplane's velocity vector is entirely in the east direction. So, its components are:
Vax = 600 mph (east component)
Vay = 0 mph (north component, which is zero because the plane is not going north or south)

The wind's velocity vector needs to be broken down into its east and north components. The angle given is measured from the south, towards the east. So, we can use sine for the east component and cosine for the north component (which will be negative because it's going south).

Wx = 85 mph * sin(59°) = 70.4 mph (east)
Wy = 85 mph * cos(59°) = -51.5 mph (south, which is negative north)

Step 2: Add the components of the vectors.

The resultant east component is the sum of the east components of the airplane and wind vectors:
Rx = Vax + Wx = 600 mph + 70.4 mph = 670.4 mph

The resultant north component is the sum of the north components of the airplane and wind vectors:
Ry = Vay + Wy = 0 mph - 51.5 mph = -51.5 mph

Step 3: Calculate the magnitude and direction of the resultant vector.

The magnitude (speed) is the square root of the sum of the squares of the components:
R = sqrt(Rx^2 + Ry^2) = sqrt((670.4 mph)^2 + (-51.5 mph)^2) = 672.3 mph

The direction is the arctangent of the north component divided by the east component. Remember to adjust the angle based on the quadrant. Since the east component is positive and the north component is negative, the angle is in the fourth quadrant:
θ = atan(Ry/Rx) = atan(-51.5 mph / 670.4 mph) = -4.4°

However, since the angle is measured from the east towards the north, we need to add 360° to the negative angle:
θ = -4.4° + 360° = 355.6°

So, the airplane's actual speed is 672.3 mph, and it's flying at an angle of 355.6° measured from the east, towards the north.

Knowee AI · Accepted Answer