An aeroplane is moving at 200km/h due 60?. The plane experience a wind blowing at 120km/h due 210?. Find the resultant velocity of the plane by resolving the velocities into their components
X-component=200*cos(60)+120cos(210)
(positive due east)
Y-component=200*sin(60)+120sin(210)
(positive due north)
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 components. The velocity of the plane is given as 200 km/h due 60?.
The horizontal component (Vx) can be found using the formula Vx = V * cos(theta), where V is the magnitude of the velocity and theta is the angle. In this case, V = 200 km/h and theta = 60?. Therefore,
Vx = 200 km/h * cos(60?)
Vx = 200 km/h * 0.5
Vx = 100 km/h
The vertical component (Vy) can be found using the formula Vy = V * sin(theta). In this case, V = 200 km/h and theta = 60?. Therefore,
Vy = 200 km/h * sin(60?)
Vy = 200 km/h * 0.866
Vy = 173.2 km/h
Now, let's resolve the velocity of the wind into its components. The velocity of the wind is given as 120 km/h due 210?.
The horizontal component (Vwx) can be found using the formula Vwx = Vw * cos(theta), where Vw is the magnitude of the wind velocity and theta is the angle. In this case, Vw = 120 km/h and theta = 210?. Therefore,
Vwx = 120 km/h * cos(210?)
Vwx = 120 km/h * (-0.866)
Vwx = -103.9 km/h
The vertical component (Vwy) can be found using the formula Vwy = Vw * sin(theta). In this case, Vw = 120 km/h and theta = 210?. Therefore,
Vwy = 120 km/h * sin(210?)
Vwy = 120 km/h * (-0.5)
Vwy = -60 km/h
Now, we can add the horizontal and vertical components of the plane's velocity with the horizontal and vertical components of the wind velocity to find the resultant velocity.
Resultant horizontal velocity (Rx) = Vx + Vwx
Rx = 100 km/h + (-103.9 km/h)
Rx = -3.9 km/h
Resultant vertical velocity (Ry) = Vy + Vwy
Ry = 173.2 km/h + (-60 km/h)
Ry = 113.2 km/h
Finally, we can calculate the magnitude of the resultant velocity (R) using the Pythagorean theorem: R = sqrt(Rx^2 + Ry^2).
R = sqrt((-3.9 km/h)^2 + (113.2 km/h)^2)
R = sqrt(15.21 km^2/h^2 + 12816.24 km^2/h^2)
R = sqrt(12831.45 km^2/h^2)
R ≈ 113.3 km/h
Therefore, the resultant velocity of the plane is approximately 113.3 km/h.
To find the resultant velocity of the plane, we need to resolve both its velocity and the wind velocity into their respective components.
Let's start by resolving the velocity of the plane:
The velocity of the plane is given as 200 km/h due 60?. This means the plane has a horizontal component and a vertical component.
To find the horizontal component, we use the equation:
Horizontal Component = Velocity * cos(angle)
Horizontal Component = 200 km/h * cos(60?)
Using the identity cos(60?) = 0.5, we can calculate:
Horizontal Component = 200 km/h * 0.5 = 100 km/h
So, the horizontal component of the velocity of the plane is 100 km/h.
To find the vertical component, we use the equation:
Vertical Component = Velocity * sin(angle)
Vertical Component = 200 km/h * sin(60?)
Using the identity sin(60?) = √3/2, we can calculate:
Vertical Component = 200 km/h * (√3/2) = 100√3 km/h
So, the vertical component of the velocity of the plane is 100√3 km/h.
Next, let's resolve the wind velocity:
The wind velocity is given as 120 km/h due 210?. Similarly, we need to find the horizontal component and the vertical component.
To find the horizontal component, we use the equation:
Horizontal Component = Velocity * cos(angle)
Horizontal Component = 120 km/h * cos(210?)
Using the identity cos(210?) = -0.866, we can calculate:
Horizontal Component = 120 km/h * (-0.866) = -104 km/h
Note that the negative sign indicates opposite direction.
So, the horizontal component of the wind velocity is -104 km/h.
To find the vertical component, we use the equation:
Vertical Component = Velocity * sin(angle)
Vertical Component = 120 km/h * sin(210?)
Using the identity sin(210?) = -0.5√3, we can calculate:
Vertical Component = 120 km/h * (-0.5√3) = -60√3 km/h
Note that the negative sign indicates opposite direction.
So, the vertical component of the wind velocity is -60√3 km/h.
Now that we have resolved both the plane's velocity and the wind's velocity into their respective components, we can find the resultant velocity by adding the corresponding components together.
Horizontal Resultant Velocity = Plane's Horizontal Component + Wind's Horizontal Component
Horizontal Resultant Velocity = 100 km/h + (-104 km/h) = -4 km/h
Vertical Resultant Velocity = Plane's Vertical Component + Wind's Vertical Component
Vertical Resultant Velocity = 100√3 km/h + (-60√3 km/h) = 40√3 km/h
So, the resultant velocity of the plane is -4 km/h horizontally and 40√3 km/h vertically.