GWANCHELE PAUL
Feb 8
A plane is moving at 200km/ h at 60°. The plane experiences a wind blowing at 120 km/h at 210°. Find resultant velocity of plane by resolving the velocities into their components ?
A plane is moving at 200km/ h at 60°. The plane experiences a wind blowing at 120 km/h at 210°. Find resultant velocity of plane by resolving the velocities into their components ?
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ChatGPT
Feb 8
To find the resultant velocity of the plane, we need to resolve the velocities into their components and then add them together.
First, let's resolve the velocity of the plane into its horizontal and vertical components. The horizontal component can be found using the cosine function:
Horizontal component of plane's velocity = 200 km/h * cos(60°)
= 200 km/h * 0.5
= 100 km/h
The vertical component can be found using the sine function:
Vertical component of plane's velocity = 200 km/h * sin(60°)
= 200 km/h * √3/2
= 100√3 km/h
Next, let's resolve the velocity of the wind into its horizontal and vertical components. The horizontal component can be found using the cosine function:
Horizontal component of wind's velocity = 120 km/h * cos(210°)
= 120 km/h * (-√3/2)
= -60√3 km/h
The negative sign indicates that the wind is blowing in the opposite direction of the positive x-axis.
The vertical component can be found using the sine function:
Vertical component of wind's velocity = 120 km/h * sin(210°)
= 120 km/h * (-1/2)
= -60 km/h
Now, let's add the horizontal components of the plane's velocity and the wind's velocity:
Resultant horizontal velocity = 100 km/h + (-60√3 km/h)
= 100 km/h - 60√3 km/h
≈ 100 km/h - 103.92 km/h
≈ -3.92 km/h
Similarly, let's add the vertical components of the plane's velocity and the wind's velocity:
Resultant vertical velocity = 100√3 km/h + (-60 km/h)
= 100√3 km/h - 60 km/h
≈ 173.21 km/h - 60 km/h
≈ 113.21 km/h
Finally, we can find the magnitude and direction of the resultant velocity using the Pythagorean theorem and inverse tangent function:
Magnitude of resultant velocity = √((-3.92 km/h)^2 + (113.21 km/h)^2)
≈ √(15.3664 km^2/h^2 + 12806.9641 km^2/h^2)
≈ √12822.3305 km^2/h^2
≈ 113.27 km/h
Direction of resultant velocity = arctan((113.21 km/h)/(-3.92 km/h))
≈ arctan(-28.85)
≈ -88.68°
Therefore, the resultant velocity of the plane is approximately 113.27 km/h at an angle of -88.68°.
First, let's resolve the velocity of the plane into its horizontal and vertical components. The horizontal component can be found using the cosine function:
Horizontal component of plane's velocity = 200 km/h * cos(60°)
= 200 km/h * 0.5
= 100 km/h
The vertical component can be found using the sine function:
Vertical component of plane's velocity = 200 km/h * sin(60°)
= 200 km/h * √3/2
= 100√3 km/h
Next, let's resolve the velocity of the wind into its horizontal and vertical components. The horizontal component can be found using the cosine function:
Horizontal component of wind's velocity = 120 km/h * cos(210°)
= 120 km/h * (-√3/2)
= -60√3 km/h
The negative sign indicates that the wind is blowing in the opposite direction of the positive x-axis.
The vertical component can be found using the sine function:
Vertical component of wind's velocity = 120 km/h * sin(210°)
= 120 km/h * (-1/2)
= -60 km/h
Now, let's add the horizontal components of the plane's velocity and the wind's velocity:
Resultant horizontal velocity = 100 km/h + (-60√3 km/h)
= 100 km/h - 60√3 km/h
≈ 100 km/h - 103.92 km/h
≈ -3.92 km/h
Similarly, let's add the vertical components of the plane's velocity and the wind's velocity:
Resultant vertical velocity = 100√3 km/h + (-60 km/h)
= 100√3 km/h - 60 km/h
≈ 173.21 km/h - 60 km/h
≈ 113.21 km/h
Finally, we can find the magnitude and direction of the resultant velocity using the Pythagorean theorem and inverse tangent function:
Magnitude of resultant velocity = √((-3.92 km/h)^2 + (113.21 km/h)^2)
≈ √(15.3664 km^2/h^2 + 12806.9641 km^2/h^2)
≈ √12822.3305 km^2/h^2
≈ 113.27 km/h
Direction of resultant velocity = arctan((113.21 km/h)/(-3.92 km/h))
≈ arctan(-28.85)
≈ -88.68°
Therefore, the resultant velocity of the plane is approximately 113.27 km/h at an angle of -88.68°.
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