An airplane is flying at an airspeed of 500 km/hr in a wind blowing at 50 km/hr toward the southeast. In what direction should the plane head to end up going due east? (Round your answer to two decimal places.)What is the airplane's speed relative to the ground? (Round your answer to the nearest whole number.)
To determine the direction the plane should head to end up going due east, we need to take into account the wind velocity.
First, let's draw a diagram to visualize the situation:
50 km/hr
Wind -----> ---------> ----------->
x = plane's airspeed (500 km/hr)
--------------- --------------->
| 500 km/hr |
--------------- --------------->
In the diagram, the arrow represents the wind blowing toward the southeast, while the line represents the plane's course.
To determine the direction, we can think of the wind speed and direction as a vector, with the length of 50 km/hr pointing southeast. We can then add the plane's airspeed vector, which has a length of 500 km/hr and is in the direction of the plane's heading.
To find the resulting velocity vector, we add the wind vector and the airspeed vector. The resulting vector will point in the direction the plane should head to end up going due east.
We can use vector addition to determine the direction:
Let's break down the wind vector into its x-component and y-component. The x-component (eastward component) of the wind vector is given by:
x-component of wind = wind speed * cos(angle)
In this case, the angle between the wind direction and due east is 45 degrees (since it's blowing toward the southeast). So,
x-component of wind = 50 km/hr * cos(45°) ≈ 50 km/hr * 0.707 ≈ 35.35 km/hr
Next, we add the x-components of the wind and airspeed vectors to determine the x-component of the resulting velocity vector:
x-component of resulting velocity = x-component of wind + x-component of airspeed
The x-component of the airspeed vector is equal to the plane's airspeed, which is 500 km/hr.
x-component of resulting velocity = 35.35 km/hr + 500 km/hr = 535.35 km/hr
The resulting velocity vector's x-component is 535.35 km/hr, which tells us that the plane should head due east.
To find the relative speed of the airplane to the ground, we can use the Pythagorean theorem. The airplane's speed relative to the ground is the magnitude of the resulting velocity vector:
Resulting velocity = √(x-component^2 + y-component^2)
The y-component of the resulting velocity vector is the y-component of the wind, which is given by:
y-component of wind = wind speed * sin(angle)
In this case, the angle between the wind direction and due east is 45 degrees.
y-component of wind = 50 km/hr * sin(45°) ≈ 50 km/hr * 0.707 ≈ 35.35 km/hr
The y-component of the resulting velocity vector is the negative of the y-component of the wind vector because the wind is blowing from the southeast.
y-component of resulting velocity = -35.35 km/hr
Now we can calculate the magnitude of the resulting velocity vector:
Resulting velocity = √(535.35 km/hr)^2 + (-35.35 km/hr)^2
Resulting velocity ≈ √(286080.22 km^2/hr^2 + 1252.14 km^2/hr^2)
Resulting velocity ≈ √287332.36 km^2/hr^2
Resulting velocity ≈ 535.99 km/hr
Therefore, the airplane's speed relative to the ground is approximately 536 km/hr.
To find the direction the plane should head to end up going due east, we need to consider the wind speed and direction. The wind is blowing toward the southeast, which means it is coming from the northwest.
To counteract the wind and fly due east, the plane should head in the opposite direction of the wind. Since the wind is coming from the northwest, the plane should head in the southeast direction.
To find the airplane's speed relative to the ground, we can use vector addition.
Let's break down the velocities:
1. Airspeed of the plane: 500 km/hr
2. Wind speed: 50 km/hr
Since the wind is blowing in the northwest direction, we can break it down into its x and y components.
Wind x-component: 50 km/hr * cos(45°)
Wind y-component: 50 km/hr * sin(45°)
By using the components, we can add the vectors to determine the airplane's speed relative to the ground.
Plane x-component: 500 km/hr - Wind x-component
Plane y-component: 0 - Wind y-component
By using vector addition, we can find the magnitude of the airplane's velocity:
Magnitude = sqrt((Plane x-component)^2 + (Plane y-component)^2)
Lastly, we round the magnitude to the nearest whole number to find the airplane's speed relative to the ground.
50 @ SE = <35.36, -35.36>
You want a direction θ such that
500 sinθ-35.36 = 0
sinθ = 35.36/500
θ = E 4.05° N