An airplane has to fly eastward to a destination 856 km away. If wind is blowing at 18.0 m/s northward and the air speed of the plane is 161 m/s, in what direction should the plane head to reach its destination?
To determine the direction the plane should head to reach its destination, we need to consider the effect of the wind. The plane's heading needs to account for both the plane's airspeed and the wind direction. We can break down the vectors involved in this situation using vector addition.
Step 1: Draw a diagram
First, draw a diagram to represent the situation. Use a vector to represent the direction and magnitude of the plane's airspeed and another vector for the direction and magnitude of the wind.
Step 2: Identify the vectors
Label the plane's airspeed vector as A and the wind vector as W. The magnitude of the plane's airspeed is 161 m/s, and the magnitude of the wind speed is 18.0 m/s.
Step 3: Resolve the vectors
We need to resolve both vectors into their eastward (x-axis) and northward (y-axis) components. Let's assume the eastward direction as positive x and the northward direction as positive y.
The plane's airspeed vector, A, can be resolved into x and y components as follows:
Ax = 161 m/s (since it is flying eastward, there is no component in the y-direction)
Ay = 0 m/s (no component in the northward direction because the plane is not trying to move in this direction)
The wind vector, W, can be resolved into x and y components as follows:
Wx = 0 m/s (no component in the eastward direction because the wind is blowing northward)
Wy = 18.0 m/s (since the wind is blowing northward)
Step 4: Add the components
Add the x and y components separately to determine the net effect of the plane's airspeed and the wind.
Net x-component:
Net_x = Ax + Wx = 161 m/s + 0 m/s = 161 m/s
Net y-component:
Net_y = Ay + Wy = 0 m/s + 18.0 m/s = 18.0 m/s
Step 5: Calculate the magnitude and direction of the resultant vector
We can now use the net x and y components to calculate the magnitude and direction of the resultant vector.
Magnitude:
Magnitude = √(Net_x^2 + Net_y^2)
= √(161^2 + 18.0^2)
= √(25921 + 324)
= √26245
≈ 161.9 m/s (rounded to one decimal place)
Direction:
To determine the direction, we can use the arctan function to find the angle. The angle is given by:
θ = arctan(Net_y/Net_x)
= arctan(18.0/161)
= arctan(0.1118)
Using a calculator, the angle θ is approximately 6.45 degrees (rounded to two decimal places).
Step 6: Conclusion
The plane should head in the direction of approximately 6.45 degrees eastward from its original heading to reach its destination.
To determine the direction the airplane should head, we need to find the resultant velocity vector by adding the vectors of the wind and the airspeed of the plane.
Let's consider the wind velocity vector as v₁ (northward) with a magnitude of 18.0 m/s and the airspeed of the plane as v₂ (eastward) with a magnitude of 161 m/s.
To find the direction the plane should head, we need to find the angle between the resultant velocity vector and the eastward direction.
First, we need to calculate the components of the wind velocity vector:
v₁x = 0 (no wind in the eastward direction)
v₁y = 18.0 m/s (northward)
Now, let's calculate the components of the resultant velocity vector:
v₂x = 161 m/s (eastward)
v₂y = 0 (no airspeed in the northward direction)
To find the resultant velocity vector, we can add the corresponding components:
vx = v₁x + v₂x = 0 + 161 = 161 m/s (eastward)
vy = v₁y + v₂y = 18.0 + 0 = 18.0 m/s (northward)
Next, we can find the magnitude and direction of the resultant velocity vector:
magnitude (v) = √(vx² + vy²) = √(161² + 18.0²) = √(25921 + 324) ≈ √26245 ≈ 161.9 m/s
The angle between the resultant velocity vector and the eastward direction can be calculated as:
θ = arctan(vy / vx) = arctan(18.0 / 161) ≈ 6.3°
Therefore, the plane should head approximately 6.3 degrees north of eastward to reach its destination.