Problem 83 - Algebra and Trigonometry

Question

The figure shows a small plane flying at a speed of 180 miles per hour on a bearing of \(\mathrm{N} 50^{\circ} \mathrm{E}\). The wind is blowing from west to east at 40 miles per hour. The figure indicates that \(\mathbf{v}\) represents the velocity of the plane in still air and w represents the velocity of the wind. a. Express \(v\) and \(w\) in terms of their magnitudes and direction angles. b. Find the resultant vector, \(\mathbf{v}+\mathbf{w}\) c. The magnitude of \(v+w\), called the ground speed of the plane, gives its speed relative to the ground. Approximate the ground speed to the nearest mile per hour. d. The direction angle of \(v+w\) gives the plane's true course relative to the ground. Approximate the true course to the nearest tenth of a degree. What is the plane's true bearing?

Step-by-Step Solution

Answer

After carrying out the calculations, we obtain the ground speed (magnitude of the resultant vector) and the true course (direction angle of the resultant vector).

Step 1: Express v and w in terms of their magnitudes and direction angles

The magnitude of \(\mathbf{v}\) is its speed which is 180 miles per hour and its direction angle is \(50^{\circ}\). Hence, its components are \(v_x = 180\cos(50^{\circ})\) and \(v_y = 180\sin(50^{\circ})\). The magnitude of the wind vector \(\mathbf{w}\) is 40 miles per hour and its direction is directly from east. Hence, its components are \(w_x = 0\) (no north or south component) and \(w_y = 40\) (eastwards).

Step 2: Find the resultant vector, v + w

The resultant vector is given by the sum of the component vectors, \(\mathbf{v}+\mathbf{w} = (v_x + w_x, v_y + w_y) = (180\cos(50^{\circ}), 180\sin(50^{\circ}) + 40)\).

Step 3: Calculate the magnitude of the resultant vector

Given that the components of the resultant vector are \((u_x, u_y)\), its magnitude is \(\sqrt{u_x^2+u_y^2}\), which yields its ground speed. Substitute the calculated components from step 2 to obtain the ground speed.

Step 4: Calculate the direction angle of the resultant vector

The direction angle \(\theta\) of the resultant vector with respect to the positive x-axis (North) can be calculated using the tangent function as \(\theta = \arctan(\frac{u_y}{u_x})\). After obtaining the angle in radians, convert it to degrees by multiplying by \(180/\pi\). Note that to obtain the real course of the plane, we need to take into account that east is referenced as \(\theta = 90^\circ\). So if the calculated \(\theta\) value is less than 90, subtract it from 90 to get the bearing angle.

Step 5: Discussion

The ground speed is the actual speed at which the plane is moving over the ground, and takes the effect of the wind into account. The true course or true bearing is the actual direction in which the plane is moving with reference to North.

Key Concepts

Velocity Components

In navigation and physics, understanding velocity components is crucial for accurately determining movement in a two-dimensional space. A velocity vector can be broken down into two perpendicular components: the horizontal and the vertical components, usually represented as \( v_x \) and \( v_y \) respectively. These components represent different directions, making it simpler to compute the overall effect of multiple forces or velocities acting on an object.

• For the plane, its velocity components in still air can be calculated using trigonometry. Given its speed (magnitude) of 180 miles per hour and direction angle of \( 50^{\circ} \), use the cosine function to find the horizontal component \( v_x = 180\cos(50^{\circ}) \) and the sine function for the vertical component \( v_y = 180\sin(50^{\circ}) \).
• The wind, blowing from west directly towards east at 40 miles per hour, contributes only to the horizontal component, resulting in \( w_x = 40 \) and \( w_y = 0 \) since there is no change in north or south direction.

This breakdown allows for the vector addition of these components, elucidating the complete impact of both velocities on the plane's movement.

Resultant Vector

When two or more vectors act upon an object, their combined effect is described by the resultant vector. This vector is found by adding the individual vectors component-wise. In the case of the plane, adding the velocity of the plane and the wind gives the resultant vector which describes the true velocity of the plane as it moves over the ground.

• The horizontal component of the resultant vector is calculated as \( v_x + w_x = 180\cos(50^{\circ}) + 40 \).
• The vertical component is given by \( v_y + w_y = 180\sin(50^{\circ}) \).

By combining these components, we get a new vector \( \mathbf{v} + \mathbf{w} = (v_x + w_x, v_y + w_y) \) which effectively provides the velocity of the plane considering the wind's influence.

Direction Angle

The direction angle is an angle that describes the orientation or path of the resultant vector relative to a reference direction, often the positive x-axis (north). This angle is pivotal in determining the true direction of travel when different forces are involved, such as wind affecting an aircraft.

• To find the direction angle \( \theta \) for the resultant vector, use the tangent function: \( \theta = \arctan\left(\frac{u_y}{u_x}\right) \), where \( u_x \) and \( u_y \) are the horizontal and vertical components of the resultant vector.
• After computing this angle, it's converted from radians to degrees using the formula: multiply by \( \frac{180}{\pi} \).
• In navigation, converting this angle to a bearing involves adjusting it relative to north, often by subtracting \( \theta \) from \( 90^{\circ} \) if \( \theta \) is less than \( 90^{\circ} \).

This enables one to get the true course, taking into account all influences on the movement of the plane.

Ground Speed

Ground speed represents the actual speed of the plane as it moves over the ground, factoring in both its airspeed and any additional factors like wind. This is calculated by determining the magnitude of the resultant vector from the velocity components.

• The formula to find the magnitude (and thus the ground speed) is \( \sqrt{u_x^2 + u_y^2} \).
• This mathematical operation sums the squares of each component, \( u_x \) and \( u_y \), derived from the resultant vector before taking the square root.
• The ground speed is cross-verified to ensure it reflects realistic conditions for air travel between two points.

By accurately calculating the ground speed, the pilot gains critical information about how fast the plane is truly traveling relative to the ground, crucial for navigation and scheduling.