Relative Motion Review— Sine and Cosine Law Solution

Summary

The video explains how to determine the heading and airspeed of a plane navigating in windy conditions. The scenario involves a pilot flying to a city 600 km away in two hours, with wind blowing at 70 km/h at 10° west of south. Through vector analysis, diagrams, and calculations using trigonometric laws (cosine and sine laws), the airspeed of 278 km/h and a heading of 1.81° south of west are determined. The process includes understanding resultant vectors and correcting for wind direction to achieve the desired flight path.

Takeaways

• 😀 The pilot's goal is to reach a city 600 km away in 2 hours, heading 15 degrees south of west.
• 🌬️ There is a 70 km/h wind blowing at 10 degrees west of south, which affects the plane's trajectory.
• 🛫 The plane's resultant velocity, considering the wind, is 300 km/h, calculated by dividing the distance by the time.
• 🔍 The script uses vector addition to determine the plane's heading and airspeed, considering the wind's effect.
• 📏 The cosine law is applied to find the plane's airspeed by using the known wind speed and resultant velocity.
• 🧮 The calculated airspeed of the plane is approximately 278 km/h, accounting for the wind's influence.
• 📐 The sine law is used to determine the angle between the wind vector and the plane's heading.
• 🔄 The plane's heading is found by subtracting the angle calculated using the sine law from the desired direction.
• 🏁 The final heading of the plane is approximately 1.81 degrees south of west, adjusted for the wind.
• 📘 The script provides a detailed step-by-step approach to solving the problem using trigonometric laws and vector analysis.

Q & A

What is the distance the pilot needs to cover to reach the city?

The pilot needs to cover a distance of 600 kilometers to reach the city.

In which direction does the pilot need to fly to reach the city?

The pilot needs to fly in a direction 15 degrees south of west to reach the city.

What is the speed and direction of the wind?

The wind is blowing at a speed of 70 kilometers per hour at a direction of 10 degrees west of south.

How long does the pilot have to reach the city?

The pilot has two hours to reach the city.

What is the resultant speed of the plane required to reach the city in the given time?

The resultant speed of the plane is 300 kilometers per hour, calculated by dividing the distance by the time.

What is the angle between the wind direction and the desired direction?

The angle between the wind direction (10 degrees west of south) and the desired direction (15 degrees south of west) is 65 degrees.

What is the formula used to calculate the unknown side of the triangle in this scenario?

The cosine law is used to calculate the unknown side of the triangle, which is represented as 'b^2 = a^2 + c^2 - 2ac cos(B)'.

What is the calculated airspeed of the plane with respect to the air?

The calculated airspeed of the plane is 278 kilometers per hour.

How is the heading of the plane determined?

The heading of the plane is determined by subtracting the angle 'a' from the desired direction angle of 15 degrees south of west.

What is the final heading of the plane after accounting for the wind?

The final heading of the plane after accounting for the wind is 1.81 degrees south of west.

What trigonometric function is used to find the angle 'a' in the triangle?

The sine function is used to find the angle 'a' in the triangle, using the formula 'sin(a) = opposite side / hypotenuse'.