an airplane flies sue west at 185 km/h with respect to the air. there is a wind blowing at 85 km/h to the northeast relative to the ground. what is the planes speed and direction with respect to the ground answers. // our class is literally stuck in this and our teacher isn't really explaining it, can someone help us?
Sure, I'd be happy to help! Let's break it down step by step.
First, we need to determine the components of the airplane's velocity. We know that the airplane flies west at 185 km/h with respect to the air. So, the westward component is simply 185 km/h.
Next, we need to determine the components of the wind's velocity. We're told that the wind is blowing at 85 km/h to the northeast relative to the ground. Since northeast is a 45-degree angle between north and east, we can split the wind velocity into its northward and eastward components. Using some simple trigonometry, we find that the northward component is 85 km/h * sin(45°) ≈ 60.1 km/h, and the eastward component is 85 km/h * cos(45°) ≈ 60.1 km/h as well.
Now, let's add up the components. Since the wind is blowing in a northeastern direction, it will partially oppose the plane's westward motion but also contribute to its northward motion. We have:
Plane's northward component: 0 km/h (no northward motion)
Plane's eastward component: -60.1 km/h (opposing the plane's westward motion)
Plane's westward component: 185 km/h (given)
So, the plane's resulting velocity with respect to the ground is obtained by adding up the components:
Resulting northward component: 0 km/h + 60.1 km/h = 60.1 km/h
Resulting eastward component: -60.1 km/h + 0 km/h = -60.1 km/h
Resulting westward component: 185 km/h
Using some more trigonometry, we can find the magnitude and direction of the plane's resulting velocity. The magnitude is given by the formula:
Resulting speed = √(Resulting northward component)^2 + (Resulting eastward component)^2
Plugging in the values:
Resulting speed = √(60.1 km/h)^2 + (-60.1 km/h)^2 ≈ 84.9 km/h
The direction is given by the formula:
Resulting direction = arctan(Resulting northward component / Resulting eastward component)
Plugging in the values:
Resulting direction = arctan(60.1 km/h / -60.1 km/h) ≈ -45°
Therefore, the plane's speed with respect to the ground is approximately 84.9 km/h, and its direction is approximately 45 degrees south of west.
I hope this helps, and remember, if anyone asks you what you did in math class, you can tell them you solved a real-life aviation mystery! Keep up the great work!
Certainly! I can help you with that problem.
To determine the plane's speed and direction with respect to the ground, we will need to break down the motion into its components. The plane's speed and direction can be represented as a vector sum of its airspeed and the wind's speed and direction.
1. First, let's find the horizontal and vertical components of the wind velocity. We know that the wind is blowing northeast, which is at a 45-degree angle between north and east. To find the component along the x-axis (horizontal component), we can use the cosine function: cos(45°) = √2/2. So, the horizontal component of the wind velocity is 85 km/h * √2/2 = 85 km/h * 0.7071 ≈ 60.21 km/h. To find the component along the y-axis (vertical component), we can use the sine function: sin(45°) = √2/2. Thus, the vertical component of the wind velocity is also 85 km/h * √2/2 = 85 km/h * 0.7071 ≈ 60.21 km/h.
2. Now, let's consider the plane's motion. It is flying west, so its velocity has a magnitude of 185 km/h in the west direction. We can represent this as (185 km/h, 270°), where 270° represents the west direction.
3. To find the plane's speed and direction with respect to the ground, we need to add the vectors of the plane's velocity and the wind's velocity. Since we have broken down the wind's velocity into horizontal and vertical components, we can simply add the corresponding components separately.
- Horizontal component: 185 km/h (planes' speed) - 60.21 km/h (wind's horizontal component) = 124.79 km/h
- Vertical component: 0 km/h (plane's vertical speed) + 60.21 km/h (wind's vertical component) = 60.21 km/h
4. Finally, we can find the resultant speed and direction using the Pythagorean theorem and trigonometry:
- Resultant speed = √(horizontal component^2 + vertical component^2) = √(124.79 km/h)^2 + (60.21 km/h)^2 ≈ √(15,584.04 + 3,625.44) ≈ √19,209.48 ≈ 138.53 km/h
- Resultant direction = arctan(vertical component / horizontal component) = arctan(60.21 km/h / 124.79 km/h) ≈ 26.78°
Therefore, the plane's speed and direction with respect to the ground is approximately 138.53 km/h at an angle of 26.78° (north of west).
Certainly! Let me explain how to solve this problem step by step.
To determine the airplane's speed and direction with respect to the ground, we need to consider the vectors involved. A vector has both magnitude (the speed) and direction.
Step 1: Draw a diagram
Start by drawing a diagram to visualize the situation. Draw a horizontal line to represent the ground, and another line to represent the motion of the airplane with a "west" arrow. Now, draw a diagonal arrow pointing from the northeast (opposite to the wind) with a length of 85 km/h.
Step 2: Break down vectors
Next, let's break down the vectors into horizontal (east-west) and vertical (north-south) components. The airplane's motion (185 km/h west) only has a horizontal component. The wind's motion (85 km/h northeast) has both horizontal and vertical components.
Step 3: Calculate vertical and horizontal components
To determine the horizontal component of the wind's motion, we use trigonometry. Since the wind is blowing northeast, we have a right triangle. The angle opposite the horizontal component is 45 degrees because northeast is a combination of north and east (which are at right angles). So, the horizontal component of the wind is 85 km/h * cos(45°).
To calculate the vertical component of the wind, use the same logic. The angle opposite the vertical component is also 45 degrees because northeast is a combination of north and east. So, the vertical component of the wind is 85 km/h * sin(45°).
Step 4: Combine horizontal components
Since the airplane is flying west, the horizontal component of the wind affects the airplane's speed. Subtract the horizontal component of the wind motion from the airplane's speed (185 km/h - horizontal component of wind).
Step 5: Combine vertical components
The vertical component of the wind doesn't affect the airplane's speed directly; it only affects the direction. Therefore, you can simply add the vertical component of the wind's motion to the airplane's vertical speed.
Step 6: Use Pythagorean theorem
To find the resultant speed with respect to the ground, use the Pythagorean theorem. The magnitude of the resultant vector is the square root of the sum of the squares of the horizontal and vertical components (found in steps 4 and 5).
Step 7: Find the resultant direction
To find the direction, you can use inverse trigonometric functions. Take the arctangent of the vertical component divided by the horizontal component (found in steps 4 and 5).
By following these steps, you can determine the airplane's speed and direction with respect to the ground.