an areroplane flies to the destination 600km west of its starting point. A wind is blowing from the North-west at 75 m/s. The pilot wants to complete the flight in 55 minutes. Calculate the direction to be taken by the pilot.
To calculate the direction the pilot should take to reach the destination, we need to consider the relative motion of the airplane and the wind.
First, let's convert the time of the flight to hours. 55 minutes is equal to 55/60 = 0.92 hours.
Given that the airplane wants to reach a destination 600 km west, we can assume that the airplane's speed relative to the ground is constant. Let's call this speed "v_a."
Now, let's break down the motion into its horizontal and vertical components.
Horizontal Motion:
The airplane needs to counteract the effects of the wind blowing from the North-west. The wind is blowing at a speed of 75 m/s, and its direction is 45 degrees counter-clockwise from the West.
We can resolve the wind speed into its horizontal and vertical components. Using trigonometry, we can calculate the horizontal component of the wind speed as follows:
Horizontal component of wind speed = Wind speed * cos(angle)
So, the horizontal component of the wind speed is:
Horizontal component = 75 m/s * cos(45 degrees) = 75 * sqrt(2)/2 ≈ 52.92 m/s
To counteract the horizontal component of the wind speed, the airplane needs to have a horizontal speed equal to the horizontal component of the wind speed.
So, the horizontal speed of the airplane, v_a, should be 52.92 m/s.
Vertical Motion:
Since we're only looking for the direction to be taken by the pilot, we can ignore the vertical motion caused by the wind.
Finally, to calculate the direction that the pilot should take, we can use trigonometry to find the angle between the airplane's velocity vector and the West direction.
Tangent of the angle = horizontal component of the airplane's velocity (v_a) divided by the vertical component (0, as we're ignoring it).
Since the vertical component is 0, we can ignore it in the calculation.
Therefore, the tangent of the required angle is:
tan(angle) = v_a / 0
tan(angle) = v_a / 0 = Undefined
As the tangent of the angle is undefined, that means the angle is either 90 degrees or 270 degrees. This is due to the fact that the airplane will need to travel either due North or due South in order to counteract the wind.
To determine which direction the pilot should take, we need more information.