A small aircraft A is about to land with an airspeed of 80 mi/hr. If the aircraft is encountering a steady side wind of speed 10 mi/hr, what angle (alpha) should the pilot direct the aircraft so the the absolute velocity is parallel to the runway? What is the speed at touchdown?

I see a lot of verbage, but nothing related to actually solving this problem.

Draw a diagram of the vectors involved. Clearly,

sin α = 10/80, so α = 7.18°

The speed at touchdown will be

80 cosα = 79.37 mi/hr

Kinematics

Average Velocity= the change in displacement of the x value time the elapsed time(avg velocity)
Vage=X/T

Average Velocity=initial velocity+final velocity divided by 2(avg velocity for constants)
Vavg=Vi+Vf/2

Acceleration=the change in velocity divided by the elapsed time
a=V/T

To find the angle (alpha) the pilot should direct the aircraft, we can use the concept of vector addition.

Let's consider two vectors: the airspeed vector (A) and the wind vector (W). The absolute velocity vector (V) would be the sum of these two vectors.

Given:
Airspeed (A) = 80 mi/hr
Side wind speed (W) = 10 mi/hr

To find the angle (alpha), we need to find the resultant vector (V) when these two vectors are added.

Using the law of vector addition, we can calculate the resultant vector (V) as follows:

V = √(A^2 + W^2 + 2 * A * W * cos(alpha))

Since we want the absolute velocity (V) to be parallel to the runway, the angle (alpha) should be the angle between the resultant vector (V) and the runway.

To make the absolute velocity parallel to the runway, the intended direction of the aircraft needs to be opposite to the wind direction, meaning it should compensate for the side wind.

Therefore, the angle (alpha) should be such that the y-component of the resultant vector (V) is equal to zero.

To calculate the speed at touchdown, we need to find the magnitude of the resultant vector (V) when the y-component is zero (V_y = 0).

Let's solve for the angle (alpha) and the speed at touchdown step-by-step:

Step 1: Set V_y = 0 and solve for alpha
0 = A * sin(alpha) + W * sin(90)
0 = A * sin(alpha) + W
A * sin(alpha) = -W
sin(alpha) = -W / A
alpha = arcsin(-W / A)

Step 2: Calculate the magnitude of the resultant vector (V) when V_y = 0
V_x = V = A * cos(alpha) + W * cos(90)
V = A * cos(alpha) + W

Step 3: Calculate the speed at touchdown
Speed at touchdown = |V|

Let's substitute the given values into these equations:

A = 80 mi/hr
W = 10 mi/hr

Step 1: Finding alpha
alpha = arcsin(-W / A)
alpha = arcsin(-10 / 80)
alpha ≈ -7.98° or -8° (rounded to the nearest degree)

Step 2: Calculating V (V_x)
V = A * cos(alpha) + W
V = 80 * cos(-8°) + 10
V ≈ 81.88 mi/hr

Step 3: Calculating the speed at touchdown
Speed at touchdown = |V|
Speed at touchdown ≈ |81.88| ≈ 81.88 mi/hr

Therefore, the pilot should direct the aircraft at an angle of approximately -8° (or -7.98°) to make the absolute velocity parallel to the runway. The speed at touchdown would be approximately 81.88 mi/hr.

To find the angle (alpha) that the pilot should direct the aircraft so that the absolute velocity is parallel to the runway, we need to break down the velocities into their components.

Let's consider the given velocities and their components:

1. Airspeed of the aircraft (v_A) = 80 mi/hr
2. Side wind velocity (v_W) = 10 mi/hr

The absolute velocity (v) can be obtained by combining the airspeed and the side wind velocity vectorially:

v = √(v_A² + v_W²)

To find the angle (alpha), we can use the following trigonometric formula:

tan(alpha) = v_W / v_A

Now let's calculate the absolute velocity and the angle alpha:

v = √(80² + 10²) = √(6400 + 100) = √(6500) ≈ 80.62 mi/hr

tan(alpha) = 10 / 80 ≈ 0.125

To find the angle alpha, we can take the inverse tangent (arctan) of 0.125:

alpha = arctan(0.125) ≈ 7.12 degrees (approximately)

Therefore, the pilot should direct the aircraft at an angle of approximately 7.12 degrees to have the absolute velocity parallel to the runway.

To find the speed at touchdown, we can use the concept of component addition.

The horizontal component of the absolute velocity (v_horizontal) will be equal to the ground speed:

v_horizontal = v_A - v_W = 80 - 10 = 70 mi/hr

Hence, the speed at touchdown will be 70 mi/hr.