AN AEROPLANE HAS TO GO FROM A POINT A TO ANOTHER POINT B, 500 KM DUE 30 DEGREE EAST OF NOTRH, A WIND IS BLOWING DUE NORTH AT A SPEED OF 20M/S. THE AIR SPEED OF THE AEROPLANE IS 150M/S.

1. FIND THE DIRECTION IN WHICH THE PILOT SHOULD HEAD THE PLANE TO REACH THE POINT B.
2. THE TIME TAKEN TO GO FROM A TO B.

To determine the direction in which the pilot should head the plane to reach point B, we need to analyze the effects of the wind on the airplane's flight path. The airplane is trying to travel 500 km at an angle of 30 degrees east of north, while a northward wind is blowing at 20 m/s. Let's break this down step by step.

Understanding the Problem

We have the following information:

• Distance from A to B: 500 km
• Direction from A to B: 30 degrees east of north
• Wind speed: 20 m/s due north
• Airplane speed: 150 m/s

Converting Units

First, we need to convert the distance from kilometers to meters for consistency with the wind speed and airplane speed:

500 km = 500,000 meters.

Setting Up the Vectors

Next, we can represent the airplane's velocity and the wind's velocity as vectors. The airplane's airspeed vector can be represented as:

• V_plane = 150 m/s at an angle of 30 degrees east of north.

To find the components of this vector, we can use trigonometry:

• V_x (eastward component) = 150 * sin(30°) = 150 * 0.5 = 75 m/s
• V_y (northward component) = 150 * cos(30°) = 150 * (√3/2) ≈ 129.9 m/s

Wind Vector

The wind is blowing due north at 20 m/s, so its vector is:

• V_wind = 0 m/s east, 20 m/s north.

Resultant Velocity

The resultant velocity of the airplane (V_resultant) can be found by adding the components of the airplane's velocity and the wind's velocity:

• V_{resultant, x} = V_{plane, x} + V_{wind, x} = 75 + 0 = 75 m/s
• V_{resultant, y} = V_{plane, y} + V_{wind, y} = 129.9 + 20 = 149.9 m/s

Finding the Required Heading

To find the direction in which the pilot should head the plane, we need to ensure that the resultant vector points directly towards point B. The angle θ of the resultant vector can be calculated using the arctangent function:

θ = arctan(V_{resultant, x} / V_{resultant, y})

θ = arctan(75 / 149.9) ≈ 26.5 degrees east of north.

This means the pilot should head approximately 26.5 degrees east of north to counteract the wind and reach point B.

Calculating the Time Taken

To find the time taken to travel from A to B, we can use the formula:

Time = Distance / Speed

Here, the effective speed towards point B is the magnitude of the resultant velocity:

Magnitude of V_resultant = √(V_{resultant, x}² + V_{resultant, y}²)

Magnitude = √(75² + 149.9²) ≈ √(5625 + 22470.01) ≈ √(28095.01) ≈ 167.7 m/s.

Now, we can calculate the time:

Time = 500,000 m / 167.7 m/s ≈ 2983.5 seconds, or approximately 49.7 minutes.

Summary

To reach point B, the pilot should head approximately 26.5 degrees east of north. The time taken to travel from A to B is about 49.7 minutes. This analysis illustrates how wind affects flight paths and the importance of vector components in navigation.