The pilot of an airplane that flies with an airspeed of 800 km/h wishes to travel to a city 800 km due east. There is a 80 km/h wind FROM the northeast

a) what should the plane's heading be?
b) How long will the trip take?

A is angle he heads North of East

plane velocity due to wind in East direction = -80 cos 45
plane velocity due to heading in East dir = 800 cos A
total East component = 800 cos A - 80 cos 45

plane velocity due to wind in North direction = -80 sin 45
plane velocity due to heading in North dir = 800 sin A

We want ZERO North resultant so
800 sin A = 80 sin 45
sin A = 0.1 (.707) = 0.0707
A = 4 degrees North of East or 90 - 4 = 86 deg compass heading

now velocity component East = 800 cos 4 - 80 cos 45
= 798 - 56.56 = 741 km/hr East
800 / 741 = hours taken

To determine the plane's heading and the duration of the trip, we need to consider the effect of the wind on the plane's ground speed. We can use vector addition to solve this problem.

Step 1: Decompose the wind velocity into its eastward and northward components.
The wind is blowing from the northeast, which is 45 degrees north of east. Using trigonometry, we can find the wind's eastward and northward components.

Eastward component of wind = wind speed * cos(angle)
= 80 km/h * cos(45°)
≈ 56.5685 km/h

Northward component of wind = wind speed * sin(angle)
= 80 km/h * sin(45°)
≈ 56.5685 km/h

Step 2: Calculate the resulting ground speed of the plane.
The ground speed is the vector sum of the plane's airspeed and the wind vector. Since the wind is blowing from the northeast, we subtract the wind vector from the plane's airspeed vector.

Eastward component of ground speed = airspeed - eastward component of wind
= 800 km/h - 56.5685 km/h
≈ 743.4315 km/h

Northward component of ground speed = northward component of wind
= 56.5685 km/h

Step 3: Calculate the magnitude and direction of the ground speed.
Using the Pythagorean theorem, we can find the magnitude of the ground speed.

Magnitude of ground speed = √[(eastward component)^2 + (northward component)^2]
= √[(743.4315 km/h)^2 + (56.5685 km/h)^2]
≈ 744.19 km/h

To find the direction of the ground speed, we use the inverse tangent function (tan⁻¹). The direction is measured in reference to due east.

Direction of ground speed = tan⁻¹(northward component / eastward component)
= tan⁻¹(56.5685 km/h / 743.4315 km/h)
≈ 4.33° north of east

a) The plane's heading should be approximately 4.33 degrees north of east.

Step 4: Calculate the duration of the trip.
To find the duration of the trip, divide the distance by the ground speed.

Duration = Distance / Ground speed
= 800 km / 744.19 km/h
≈ 1.0755 hours

b) The trip will take approximately 1.08 hours (or 1 hour and 5 minutes).

To find the answers to these questions, we need to break down the problem into different components and analyze each one individually. Let's start by determining the plane's true airspeed and groundspeed.

a) To find the plane's heading, we need to calculate the angle at which it should fly. This can be determined by considering the effect of the wind.

Since the wind is coming from the northeast, it creates a right-angled triangle with the plane's heading. The wind's direction is 45 degrees (north-east), and the angle between the wind and the plane's heading is 90 degrees.

To find the plane's heading, we can take the tangent of the angle between the heading and the wind. The angle can be calculated as arctan(80/800) = 5.7 degrees (approximately).

Therefore, the plane's heading should be 5.7 degrees north of due east.

b) To calculate the duration of the trip, we'll need to find the groundspeed of the plane. The groundspeed is the plane's true airspeed plus or minus the wind speed depending on whether the wind is helping or hindering the plane's motion.

In this case, the wind is opposing the plane's motion, so we need to subtract the wind speed from the airspeed to determine the groundspeed.

Groundspeed = Airspeed - Wind Speed
Groundspeed = 800 km/h - 80 km/h = 720 km/h

Now that we have the groundspeed, we can calculate the time taken for the trip.

Time = Distance / Speed
Time = 800 km / 720 km/h ≈ 1.1 hours

Therefore, the trip will take approximately 1.1 hours.