The captain of a plane wishes to proceed due west. The
cruising speed of the plane is 245 m/s relative to the air. A
weather report indicates that a 38.0 m/s wind is blowing
from south to north. In what direction should the pilot head
the plane relative to the ground?
Vpw = Vp + Vw = -245 m/s
Vp + 38i = -245
Vp = -245-38i(Q3).
Tan A = Y/X = -38/-245 = 0.15510
A = 8.82o S. of W. = 188.82o CCW.
To determine the direction in which the pilot should head the plane relative to the ground, we need to consider the vector addition of the plane's velocity and the wind's velocity.
Let's break down the velocities into their respective components:
The plane's velocity: 245 m/s due west
Wind's velocity: 38.0 m/s from south to north
Next, we need to find the resultant velocity by adding these two vectors. We can do this by using vector addition.
The west component of the plane's velocity is 245 m/s, and the north component of the wind's velocity is 38.0 m/s. Since these components are perpendicular to each other, we can directly add them to get the north component of the resultant velocity.
245 m/s (west) + 38.0 m/s (north) = 283.0 m/s (north-west)
Similarly, the south component of the wind's velocity is 38.0 m/s, and since it is opposite in direction to the plane's velocity, we subtract it from the west component of the plane's velocity.
245 m/s (west) - 38.0 m/s (south) = 207.0 m/s (west)
Now that we have the east-west and north-south components of the resultant velocity, we can use the tangent function to find the direction. The direction is given by the inverse tangent (tan⁻¹) of the north component divided by the east component.
Direction = tan⁻¹ (north component / west component)
Direction = tan⁻¹ (283.0 m/s / 207.0 m/s)
Using a calculator, the direction is approximately 52.2° north-west of west.
Therefore, the pilot should head the plane approximately 52.2° north-west of west relative to the ground.
To determine the direction the pilot should head the plane relative to the ground, we need to consider the effect of the wind.
Step 1: Draw a diagram:
Start by drawing a diagram that represents the direction of the wind, the plane's heading, and the resulting ground velocity.
Step 2: Decompose the velocities:
Break down the velocities into their horizontal (east-west) and vertical (north-south) components.
The plane's cruising speed relative to the air is 245 m/s due west (east-west component: -245 m/s, north-south component: 0 m/s).
The wind is blowing from south to north at 38.0 m/s (east-west component: 0 m/s, north-south component: 38.0 m/s).
Step 3: Calculate the resulting velocity:
Add the horizontal and vertical components of the plane's velocity and the wind's velocity together to get the resulting velocity.
East-West component: -245 m/s + 0 m/s = -245 m/s
North-South component: 0 m/s + 38.0 m/s = 38.0 m/s
Step 4: Find the direction of the resulting velocity:
Use the trigonometric function tangent to find the angle of the resulting velocity.
tanθ = North-South component / East-West component
θ = arctan(38.0 m/s / -245 m/s)
Using a calculator, we find that θ is approximately -8.9°.
Step 5: Determine the direction for the pilot to head the plane:
The negative angle indicates a direction to the left (west) of due west.
Therefore, the pilot should head the plane approximately 8.9° west of due west to counteract the effect of the wind and maintain a westward course relative to the ground.