A pilot must fly his plane to a town which is 200km from his starting point in a direction 30degrees N of E. He must make the trip in 1/2 h. An 80km/h wind blows in a direction 30degrees E of S. Find the speed of the plane relative to air.
I found the speed of the plane relative to the ground is 400km/h.
The answer for the speed of the plane relative to the air is 408 km/h. Can someone please explain HOW you would get that. I've spent much too mlong trying to figure it out myself and it just doesnt make any sense. Thanks so much
To find the speed of the plane relative to the air, we need to consider the effect of the wind on the plane's motion. We can break down the problem into two components: the plane's eastward motion and its northward motion.
First, let's find the eastward velocity component of the wind. The wind is blowing in a direction 30 degrees E of S, which is 30 degrees clockwise from the south direction. Using trigonometry, we can find the eastward component of the wind velocity as follows:
Wind velocity (east) = wind speed * cos(angle)
Wind velocity (east) = 80 km/h * cos(30 degrees)
Wind velocity (east) = 80 km/h * (√3/2)
Wind velocity (east) ≈ 69.282 km/h
Next, let's find the northward velocity component of the wind:
Wind velocity (north) = wind speed * sin(angle)
Wind velocity (north) = 80 km/h * sin(30 degrees)
Wind velocity (north) = 80 km/h * (1/2)
Wind velocity (north) = 40 km/h
Now, let's consider the plane's motion. It needs to travel a distance of 200 km in 1/2 hour, which means its eastward velocity is:
Plane velocity (east) = eastward distance / time
Plane velocity (east) = 200 km / (1/2) h
Plane velocity (east) = 200 km/h * 2
Plane velocity (east) = 400 km/h
Now, let's consider the northward component of the plane's motion. Since the plane is flying in a direction 30 degrees N of E, we can find the northward velocity as follows:
Plane velocity (north) = plane velocity * sin(angle)
Plane velocity (north) = 400 km/h * sin(30 degrees)
Plane velocity (north) = 400 km/h * (1/2)
Plane velocity (north) = 200 km/h
Now, we can find the combined effect of the wind and the plane's motion. The resultant eastward velocity is the difference between the plane's eastward velocity and the wind's eastward velocity:
Resultant velocity (east) = Plane velocity (east) - Wind velocity (east)
Resultant velocity (east) = 400 km/h - 69.282 km/h
Resultant velocity (east) ≈ 330.718 km/h
Similarly, the resultant northward velocity is the sum of the plane's northward velocity and the wind's northward velocity:
Resultant velocity (north) = Plane velocity (north) + Wind velocity (north)
Resultant velocity (north) = 200 km/h + 40 km/h
Resultant velocity (north) = 240 km/h
Now, we can use these resultant velocities to find the magnitude of the resultant velocity vector:
Resultant velocity = √((resultant velocity (east))^2 + (resultant velocity (north))^2)
Resultant velocity = √((330.718 km/h)^2 + (240 km/h)^2)
Resultant velocity ≈ √(109229.822564 + 57600)
Resultant velocity ≈ √166829.822564
Resultant velocity ≈ 408.128 km/h
Therefore, the speed of the plane relative to the air is approximately 408 km/h.