An airplane is going 300 mph at a bearing of N 55 degrees E. The wind is blowing 30 mph due East. What is the actual speed and direction of the plane?

V = 300mph[35o] + 30mph[0o]

X = 300*cos35+30 = 275.7 mph.
Y = 300*sin35 = 172.1 mph.

tanA = 172.1/275.7 = 0.62413
A = 32o

V = X/cosA = 275.7/cos32=325.1 mph[32o]

To find the actual speed and direction of the plane, we need to consider the effect of the wind on the plane's motion.

First, let's analyze the components of the plane's velocity and the wind velocity.

The plane's velocity can be broken down into two components: the northward component and the eastward component. We can use trigonometry to find these components.

The northward component of the plane's velocity is given by:
northward component = plane's speed * sin(bearing angle) = 300 mph * sin(55 degrees) ≈ 243.38 mph

The eastward component of the plane's velocity is given by:
eastward component = plane's speed * cos(bearing angle) = 300 mph * cos(55 degrees) ≈ 191.21 mph

The wind velocity is purely in the eastward direction, with a magnitude of 30 mph.

Next, let's add the wind's effect to the plane's velocity.

The northward component of the new velocity is still 243.38 mph as the wind blows purely eastward and doesn't affect the northern direction.

To find the eastward component of the new velocity, we add the eastward component of the plane's initial velocity and the wind velocity:
eastward component = 191.21 mph (plane's eastward velocity) + 30 mph (wind's eastward velocity) = 221.21 mph

The actual speed of the plane (magnitude of the new velocity vector) can be found using the Pythagorean theorem:
actual speed = sqrt(northward component^2 + eastward component^2)
= sqrt(243.38 mph^2 + 221.21 mph^2)
≈ sqrt(59351.7844 + 48912.0841)
≈ sqrt(108263.8685)
≈ 329.09 mph

Lastly, the direction of the plane can be found using inverse trigonometry. The angle is given by the eastward component divided by the northward component:
direction = arctan(eastward component / northward component)
= arctan(221.21 mph / 243.38 mph)
≈ arctan(0.9077)
≈ 43.79 degrees

Therefore, the actual speed of the plane is approximately 329.09 mph, and the direction is N 43.79 degrees E.

To determine the actual speed and direction of the plane, we need to use vector addition. Here's how you can solve the problem step by step:

Step 1: Start by breaking the airplane's velocity into its North (N) and East (E) components.

The bearing, N 55 degrees E, means that the angle between the plane's velocity and the North direction is 55 degrees. To find the North component, we can use Trigonometry.

North Component = Plane's Speed * Cos(angle)
North Component = 300 mph * Cos(55 degrees)
North Component ≈ 174.17 mph

To find the East component, we can use Trigonometry again.

East Component = Plane's Speed * Sin(angle)
East Component = 300 mph * Sin(55 degrees)
East Component ≈ 244.54 mph

Step 2: Determine the wind's velocity components.

Given that the wind is blowing due East at 30 mph, its velocity can be expressed as (0 mph, 30 mph) in North and East components.

Step 3: Calculate the resulting velocity.

To find the actual speed and direction of the plane, we need to add the plane's velocity components to the wind's velocity components.

North Resultant = Plane's North Component + Wind's North Component
North Resultant = 174.17 mph + 0 mph
North Resultant ≈ 174.17 mph

East Resultant = Plane's East Component + Wind's East Component
East Resultant = 244.54 mph + 30 mph
East Resultant ≈ 274.54 mph

Step 4: Use the resulting velocity components to calculate the actual speed and direction.

The actual speed can be determined using the Pythagorean theorem, as follows:

Actual Speed = √(North Resultant^2 + East Resultant^2)
Actual Speed = √(174.17 mph)^2 + (274.54 mph)^2)
Actual Speed ≈ 327.40 mph

To find the direction, we can use Trigonometry. The angle between the actual velocity and the North direction can be found as follows:

Angle = Tan^(-1)(East Resultant / North Resultant)
Angle = Tan^(-1)(274.54 mph / 174.17 mph)
Angle ≈ 58.29 degrees

Therefore, the actual speed of the plane is approximately 327.40 mph, and the direction is N 58.29 degrees E.