The nose of an ultralight plane is pointed due south, and its airspeed indicator shows 35 m/s. The plane is in a 10–m/s wind blowing toward the southwest relative to the earth. (b) Let x be east and y be north, and find the components of υ→ P/E.

Verified step by step guidance

1. Define the coordinate system where x is east and y is north. The plane's velocity relative to the air (υ→ P/A) is 35 m/s due south, which can be represented as (0, -35) m/s.
2. The wind's velocity relative to the earth (υ→ W/E) is 10 m/s toward the southwest. Since southwest is equally south and west, decompose this velocity into its x and y components. Each component will be -10/√2 m/s in the x-direction (west) and -10/√2 m/s in the y-direction (south).
3. Use vector addition to find the plane's velocity relative to the earth (υ→ P/E). This is done by adding the plane's velocity relative to the air (υ→ P/A) and the wind's velocity relative to the earth (υ→ W/E).
4. Calculate the x-component of υ→ P/E: Add the x-components of υ→ P/A and υ→ W/E. Since υ→ P/A has an x-component of 0 m/s, the x-component of υ→ P/E will be just the x-component of υ→ W/E.
5. Calculate the y-component of υ→ P/E: Add the y-components of υ→ P/A and υ→ W/E. Sum the y-component of υ→ P/A (-35 m/s) and the y-component of υ→ W/E (-10/√2 m/s).

Key Concepts

Vector Components

Vector components are the projections of a vector along the axes of a coordinate system. In this context, the plane's velocity and wind velocity can be broken down into their east-west (x) and north-south (y) components. This allows for easier calculations of the resultant velocity by combining these components using vector addition.

Relative Velocity

Relative velocity refers to the velocity of an object as observed from a particular reference frame. In this problem, the plane's velocity is affected by the wind's velocity, which is also defined relative to the earth. Understanding how to combine these velocities is crucial for determining the plane's actual motion relative to the ground.

Wind Vector

A wind vector represents the speed and direction of the wind. In this scenario, the wind is blowing toward the southwest at 10 m/s, which can be expressed in terms of its x and y components. This is essential for calculating the resultant velocity of the plane, as the wind's influence must be accounted for in the overall motion.