The velocity of an aeroplane in still air is 800km/h on a bearing of 060°. If the wind is blowing at 100km/h on a bearing of 300°,find the actual speed and direction of the plane.
To find the actual speed and direction of the plane, we need to find the resultant velocity vector by adding the velocity of the plane and the velocity of the wind vectors.
The velocity of the plane is given by:
Velocity of plane = 800 km/h at a bearing of 060°
The velocity of the wind is given by:
Velocity of wind = 100 km/h at a bearing of 300°
To add these vectors, we need to break them down into their horizontal and vertical components.
For the velocity of the plane:
Horizontal component = Velocity of plane x cos(bearing) = 800 km/h x cos(60°) = 800 km/h x 0.5 = 400 km/h
Vertical component = Velocity of plane x sin(bearing) = 800 km/h x sin(60°) = 800 km/h x 0.866 = 692.8 km/h
For the velocity of the wind:
Horizontal component = Velocity of wind x cos(bearing) = 100 km/h x cos(300°) = 100 km/h x (-0.866) = -86.6 km/h
Vertical component = Velocity of wind x sin(bearing) = 100 km/h x sin(300°) = 100 km/h x (-0.5) = -50 km/h
Now, we can add the horizontal and vertical components separately to get the resultant velocity vector:
Horizontal component of resultant velocity = Horizontal component of plane + Horizontal component of wind = 400 km/h + (-86.6 km/h) = 313.4 km/h
Vertical component of resultant velocity = Vertical component of plane + Vertical component of wind = 692.8 km/h + (-50 km/h) = 642.8 km/h
To find the magnitude (actual speed) and direction of the resultant velocity vector, we can use the Pythagorean theorem and inverse tangent function:
Magnitude of resultant velocity = √((Horizontal component of resultant velocity)^2 + (Vertical component of resultant velocity)^2) = √(313.4^2 + 642.8^2) ≈ 718.4 km/h
Direction of resultant velocity = atan(Vertical component of resultant velocity / Horizontal component of resultant velocity) = atan(642.8 km/h / 313.4 km/h) ≈ 1.14 rad
Converting the direction from radians to degrees:
Direction of resultant velocity ≈ 65.49°
Therefore, the actual speed of the plane is approximately 718.4 km/h, and its direction is approximately 65.49°.
To find the actual speed and direction of the plane, we need to calculate the resultant vector of the velocity of the plane and the velocity of the wind.
Step 1: Convert the velocities to Cartesian coordinates.
The velocity of the plane in still air can be represented as (-800 cos(60°), 800 sin(60°)) km/h.
The velocity of the wind can be represented as (-100 cos(300°), -100 sin(300°)) km/h.
Step 2: Add the x and y components.
Adding the x-components, we get:
-800 cos(60°) + (-100 cos(300°)) km/h
Simplifying:
-800 * 0.5 + (-100 * 0.866) km/h
-400 - 86.6 km/h
-486.6 km/h
Adding the y-components, we get:
800 sin(60°) + (-100 sin(300°)) km/h
Simplifying:
800 * 0.866 + (-100 * (-0.5)) km/h
692.8 + 50 km/h
742.8 km/h
Step 3: Calculate the magnitude and direction of the resultant vector.
The magnitude of the resultant vector is given by:
√((-486.6)^2 + (742.8)^2) km/h
√(236169.56 + 551180.84) km/h
√(787350.4) km/h
887.3 km/h (rounded to one decimal place)
The direction of the resultant vector is given by:
tan^(-1)(742.8 / -486.6)
tan^(-1)(-1.526)
-57.5° (rounded to one decimal place)
Therefore, the actual speed of the plane is approximately 887.3 km/h on a bearing of 057.5°.