GWANCHELE PAUL
Feb 8
A plane is flying at a speed of 300km/h relative t ground at 300° from east. The flying amidst a wind blowing at 85km/h at 225° from east . What is velocity of plane relative to air?
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ChatGPT
Feb 8
To find the velocity of the plane relative to the air, we need to break down the velocities into their horizontal and vertical components.
The velocity of the plane relative to the ground can be broken down into its horizontal and vertical components using trigonometry. The horizontal component is given by the formula:
horizontal component = velocity * cos(angle)
In this case, the angle is 300° from east, so the horizontal component of the velocity of the plane relative to the ground is:
horizontal component = 300 km/h * cos(300°) = 300 km/h * cos(60°) = 300 km/h * 0.5 = 150 km/h
The vertical component is given by the formula:
vertical component = velocity * sin(angle)
In this case, the angle is 300° from east, so the vertical component of the velocity of the plane relative to the ground is:
vertical component = 300 km/h * sin(300°) = 300 km/h * sin(60°) = 300 km/h * √3/2 = 150 km/h * √3
Similarly, we can break down the velocity of the wind into its horizontal and vertical components. The horizontal component is given by the formula:
horizontal component = velocity * cos(angle)
In this case, the angle is 225° from east, so the horizontal component of the velocity of the wind is:
horizontal component = 85 km/h * cos(225°) = 85 km/h * cos(45°) = 85 km/h * √2/2 = 42.5 km/h * √2
The vertical component is given by the formula:
vertical component = velocity * sin(angle)
In this case, the angle is 225° from east, so the vertical component of the velocity of the wind is:
vertical component = 85 km/h * sin(225°) = 85 km/h * sin(45°) = 85 km/h * √2/2 = 42.5 km/h * √2
To find the velocity of the plane relative to the air, we subtract the velocity of the wind from the velocity of the plane relative to the ground:
horizontal component of velocity of plane relative to air = horizontal component of velocity of plane relative to ground - horizontal component of velocity of wind
= 150 km/h - 42.5 km/h * √2
vertical component of velocity of plane relative to air = vertical component of velocity of plane relative to ground - vertical component of velocity of wind
= 150 km/h * √3 - 42.5 km/h * √2
The magnitude of the velocity of the plane relative to the air is given by the formula:
velocity = √(horizontal component^2 + vertical component^2)
Substituting the values we found earlier:
velocity = √((150 km/h - 42.5 km/h * √2)^2 + (150 km/h * √3 - 42.5 km/h * √2)^2)
Calculating this expression will give us the velocity of the plane relative to the air.
The velocity of the plane relative to the ground can be broken down into its horizontal and vertical components using trigonometry. The horizontal component is given by the formula:
horizontal component = velocity * cos(angle)
In this case, the angle is 300° from east, so the horizontal component of the velocity of the plane relative to the ground is:
horizontal component = 300 km/h * cos(300°) = 300 km/h * cos(60°) = 300 km/h * 0.5 = 150 km/h
The vertical component is given by the formula:
vertical component = velocity * sin(angle)
In this case, the angle is 300° from east, so the vertical component of the velocity of the plane relative to the ground is:
vertical component = 300 km/h * sin(300°) = 300 km/h * sin(60°) = 300 km/h * √3/2 = 150 km/h * √3
Similarly, we can break down the velocity of the wind into its horizontal and vertical components. The horizontal component is given by the formula:
horizontal component = velocity * cos(angle)
In this case, the angle is 225° from east, so the horizontal component of the velocity of the wind is:
horizontal component = 85 km/h * cos(225°) = 85 km/h * cos(45°) = 85 km/h * √2/2 = 42.5 km/h * √2
The vertical component is given by the formula:
vertical component = velocity * sin(angle)
In this case, the angle is 225° from east, so the vertical component of the velocity of the wind is:
vertical component = 85 km/h * sin(225°) = 85 km/h * sin(45°) = 85 km/h * √2/2 = 42.5 km/h * √2
To find the velocity of the plane relative to the air, we subtract the velocity of the wind from the velocity of the plane relative to the ground:
horizontal component of velocity of plane relative to air = horizontal component of velocity of plane relative to ground - horizontal component of velocity of wind
= 150 km/h - 42.5 km/h * √2
vertical component of velocity of plane relative to air = vertical component of velocity of plane relative to ground - vertical component of velocity of wind
= 150 km/h * √3 - 42.5 km/h * √2
The magnitude of the velocity of the plane relative to the air is given by the formula:
velocity = √(horizontal component^2 + vertical component^2)
Substituting the values we found earlier:
velocity = √((150 km/h - 42.5 km/h * √2)^2 + (150 km/h * √3 - 42.5 km/h * √2)^2)
Calculating this expression will give us the velocity of the plane relative to the air.
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