Understanding the Path of an Aeroplane

An aeroplane travels in a series of segments, each with its own direction and distance. By combining these segments using vector addition and trigonometric functions, we can determine the final position of the plane relative to its starting point. This process involves breaking down each segment into its horizontal (east) and vertical (north) components and then summing these components to find the overall displacement.

Breaking Down Each Segment

Let's consider the journey of an aeroplane that first flies 20 km in a direction 60° north of east, then 30 km straight east, and finally 10 km straight north. We can break this complex journey into simpler, segment-wise calculations:

First Segment: 20 km at 60° north of east Second Segment: 30 km straight east Third Segment: 10 km straight north

Each segment can be broken down into its respective east and north components.

First Segment: 20 km at 60° north of east

- East component: 20 cos(60°) 20 * 0.5 10 km - North component: 20 sin(60°) 20 * (sqrt(3)/2) ≈ 17.32 km

Second Segment: 30 km straight east

- East component: 30 km - North component: 0 km

Third Segment: 10 km straight north

- East component: 0 km - North component: 10 km

Summing the Components

Now, let's sum the east and north components separately to find the total displacement in each direction:

- Total East Component: 10 km 30 km 0 km 40 km - Total North Component: 17.32 km 0 km 10 km 27.32 km

With the total east and north components, we can now calculate the actual distance of the aeroplane from the starting point using the Pythagorean theorem:

[text{Distance} sqrt{40^2 27.32^2} approx sqrt{1600 745.3584} approx sqrt{2345.3584} approx 48.43 text{ km}]

Calculating the Direction

Finally, to find the direction in which the aeroplane is from its starting point, we use the tangent function:

[tan(theta) frac{text{Total North Component}}{text{Total East Component}} frac{27.32}{40}] [theta tan^{-1} left(frac{27.32}{40}right) approx tan^{-1} (0.683) approx 34.49^{circ}]

This angle is measured from the east in the direction towards the north.

Conclusion

In summary, after traveling the described path, the aeroplane is approximately 48.43 km from the starting point, at an angle of 34.49o north of east. This method of using vector addition and trigonometry is a powerful tool in navigation and can be applied to various real-world scenarios, from tracking the movement of ships at sea to determining the position of satellites in orbit.