In still air a plane has air speed 300km/h and flies on a bearing of 110 degrees. If the wind is blowing at a speed of 100km/h from a bearing of 60 degrees, find the actual velocity of the plane.
Add these two vectors:
(1) plane velocity in still air and (2) wind speed vector
To find the actual velocity of the plane, we need to consider both the air speed of the plane and the effect of the wind on its motion. We can break down the air speed and wind velocity into their horizontal and vertical components and then determine their resultant horizontal and vertical velocities.
Step 1: Resolve the air speed and wind velocity into their horizontal and vertical components.
- The air speed of the plane is 300 km/h and it is flying on a bearing of 110 degrees.
To find the horizontal component, we use the formula: horizontal component = air speed * cos(bearing)
Therefore, horizontal component of the air speed = 300 km/h * cos(110 degrees) = -134.77 km/h (negative indicates westward direction)
To find the vertical component, we use the formula: vertical component = air speed * sin(bearing)
Therefore, vertical component of the air speed = 300 km/h * sin(110 degrees) = 267.95 km/h (positive indicates northward direction)
- The wind speed is 100 km/h and it is blowing from a bearing of 60 degrees.
To find the horizontal component, we use the same formula: horizontal component = wind speed * cos(bearing)
Therefore, horizontal component of the wind speed = 100 km/h * cos(60 degrees) = 50 km/h (positive indicates eastward direction)
To find the vertical component, we use the same formula: vertical component = wind speed * sin(bearing)
Therefore, vertical component of the wind speed = 100 km/h * sin(60 degrees) = 86.60 km/h (positive indicates northward direction)
Step 2: Determine the resultant horizontal and vertical velocities.
To find the resultant horizontal velocity, we need to add the horizontal components of the air speed and wind speed together.
Resultant horizontal velocity = horizontal component of air speed + horizontal component of wind speed
Resultant horizontal velocity = -134.77 km/h + 50 km/h = -84.77 km/h (negative indicates westward direction)
To find the resultant vertical velocity, we need to add the vertical components of the air speed and wind speed together.
Resultant vertical velocity = vertical component of air speed + vertical component of wind speed
Resultant vertical velocity = 267.95 km/h + 86.60 km/h = 354.55 km/h (positive indicates northward direction)
Step 3: Calculate the magnitude of the actual velocity.
To find the magnitude of the actual velocity, we can use the Pythagorean theorem.
Magnitude of actual velocity = square root of (resultant horizontal velocity^2 + resultant vertical velocity^2)
Magnitude of actual velocity = square root of ((-84.77 km/h)^2 + (354.55 km/h)^2)
Magnitude of actual velocity = square root of (7186.29 + 125631.03) = square root of 132817.32
Magnitude of actual velocity ≈ 364.52 km/h
Therefore, the magnitude of the actual velocity of the plane is approximately 364.52 km/h.