An airplane is moving at 200km/h at an angle of 60 degree. The plane experience a wind blowing 120km/h at an angle of 210 degree. Find the resultant velocity of plane by resolving the velocities into their components
To find the resultant velocity of the plane, we need to resolve the velocities into their components and then add them together.
Let's first resolve the velocity of the plane into its components. The component in the x-direction can be found using the formula:
Vx = V * cos(θ)
where V is the magnitude of the velocity (200 km/h) and θ is the angle (60 degrees). Plugging in the values:
Vx = 200 km/h * cos(60 degrees) = 200 km/h * 0.5 = 100 km/h
The component in the y-direction can be found using the formula:
Vy = V * sin(θ)
Plugging in the values:
Vy = 200 km/h * sin(60 degrees) = 200 km/h * √3/2 = 100√3 km/h
Now let's resolve the velocity of the wind into its components. The component in the x-direction can be found using the formula:
Vwx = Vw * cos(θw)
where Vw is the magnitude of the wind velocity (120 km/h) and θw is the angle (210 degrees). Plugging in the values:
Vwx = 120 km/h * cos(210 degrees) = 120 km/h * (-0.5) = -60 km/h
The negative sign indicates that the wind is blowing in the opposite direction of the positive x-axis.
The component in the y-direction can be found using the formula:
Vwy = Vw * sin(θw)
Plugging in the values:
Vwy = 120 km/h * sin(210 degrees) = 120 km/h * (-√3/2) = -60√3 km/h
Now let's add the components of the plane's velocity and the wind's velocity to find the resultant velocity.
The x-component of the resultant velocity is:
Vrx = Vx + Vwx = 100 km/h + (-60 km/h) = 40 km/h
The y-component of the resultant velocity is:
Vry = Vy + Vwy = 100√3 km/h + (-60√3 km/h) = 40√3 km/h
Finally, we can find the magnitude of the resultant velocity using the Pythagorean theorem:
Resultant velocity = √(Vrx^2 + Vry^2) = √(40 km/h)^2 + (40√3 km/h)^2 = √(1600 km/h^2 + 4800 km/h^2) = √(6400 km/h^2) = 80 km/h
Therefore, the resultant velocity of the plane is 80 km/h.