Problem 128 - Fundamentals of Physics

Question

The pilot of an aircraft flies due east relative to the ground in a wind blowing \(20.0 \mathrm{~km} / \mathrm{h}\) toward the south. If the speed of the aircraft in the absence of wind is \(70.0 \mathrm{~km} / \mathrm{h}\), what is the speed of the aircraft relative to the ground?

Step-by-Step Solution

Answer

The speed of the aircraft relative to the ground is approximately 72.8 km/h.

Step 1: Identify Components

First, identify the components of the aircraft's velocity and the wind velocity. The aircraft's speed in still air is 70.0 km/h east, and the wind's speed is 20.0 km/h south.

Step 2: Represent as Vectors

Represent the aircraft's velocity as the vector \( \mathbf{V_a} = 70.0 \ \hat{i} \) km/h and the wind's velocity as the vector \( \mathbf{V_w} = 20.0 \ \hat{j} \) km/h, where \( \hat{i} \) and \( \hat{j} \) are unit vectors in the east and south directions, respectively.

Step 3: Calculate Resultant Velocity

The resultant velocity of the aircraft relative to the ground is the vector sum of the aircraft's velocity and wind velocity. Thus, \( \mathbf{V_r} = 70.0 \ \hat{i} + 20.0 \ \hat{j} \).

Step 4: Compute Magnitude of Resultant Velocity

The magnitude of the resultant velocity \( \|\mathbf{V_r}\| \) is calculated using the Pythagorean theorem: \[ \|\mathbf{V_r}\| = \sqrt{(70.0)^2 + (20.0)^2} \]

Step 5: Solve for Speed

Calculate the magnitude: \[ \|\mathbf{V_r}\| = \sqrt{4900 + 400} = \sqrt{5300} \approx 72.8 \text{ km/h} \].

Key Concepts

Vector Addition

Understanding vector addition is crucial in determining the resultant velocity of the aircraft relative to the ground. When we talk about vectors in physics, we are referring to quantities that have both a magnitude and a direction. In this problem, the aircraft's velocity and the wind's velocity are represented as vectors.

• The aircraft’s vector, traveling east, is given a positive value on the horizontal (x-axis), represented by the unit vector \( \hat{i} \).
• The wind’s vector, blowing south, is represented on the vertical (y-axis) with the unit vector \( \hat{j} \), typically negative due to its direction opposite to the north.

To find the speed of the aircraft relative to the ground, we perform vector addition, combining the vectors of the aircraft and the wind. This is done by adding their corresponding components: the eastward component from the aircraft and the southward component from the wind. The result is a new vector that reflects how fast and in what direction the aircraft is moving across the ground. This process uses the simple vector addition formula:\[\mathbf{V_r} = \mathbf{V_a} + \mathbf{V_w} = 70.0 \, \hat{i} + 20.0 \, \hat{j}\].

Pythagorean Theorem

The Pythagorean theorem is key in calculating the magnitude of the resultant vector obtained from vector addition. This theorem applies to right-angled triangles, positing that the square of the hypotenuse is equal to the sum of the squares of the other two sides. In our context, we apply this to calculate the resultant speed of the aircraft relative to the ground.

When the vectors \( \mathbf{V_a} \) (70.0 km/h east) and \( \mathbf{V_w} \) (20.0 km/h south) are added, they form a right triangle.

• The eastward velocity acts as one side.
• The southward velocity forms the other side.
• The resultant vector \( \mathbf{V_r} \) represents the hypotenuse.

Thus, to find the speed of the aircraft relative to the ground, we compute the magnitude of the resultant by employing the Pythagorean theorem formula:\[\|\mathbf{V_r}\| = \sqrt{(70.0 \, \text{km/h})^2 + (20.0 \, \text{km/h})^2}\]Upon solving the above, \[\|\mathbf{V_r}\| = \sqrt{4900 + 400} = \sqrt{5300} \approx 72.8 \, \text{km/h}\] This shows us how much and how swiftly the aircraft travels relative to the ground.

Aircraft Navigation

Navigating an aircraft involves understanding various factors affecting its course and speed, including wind velocity. Pilots must account for the wind when planning a flight path to ensure the aircraft reaches its intended destination efficiently.

In this problem, the pilot must consider the southward wind pushing against the aircraft. The goal is to calculate the correct heading and airspeed so that the actual path taken by the aircraft over the ground matches the planned route.

• We blend navigational experience with physics and mathematics.
• Accurate navigation ensures minimal fuel consumption and on-time arrivals.
• Pilots use instruments and calculations, like vector addition, to reach this goal.

By understanding how to manage these vectors and wind conditions, pilots can effectively control the direction and speed of the aircraft for safe and precise arrival at the destination.