A plane is flying at a velocity of 300 km/h relative to the ground in the direction 300° from East. The plane is flying amidst a wind blowing at 85 km/h relative to the ground in the direction 225° from East. What is the velocity of the plane relative to the air?
To solve this problem, we can break down the velocity of the plane and the wind into their horizontal and vertical components.
The velocity of the plane relative to the ground is given as 300 km/h in the direction 300° from East. We can break this down into horizontal (x) and vertical (y) components using trigonometry.
The horizontal component of the plane's velocity is given by:
x = 300 km/h * cos(300°) = 300 km/h * cos(60°) = 300 km/h * 0.5 = 150 km/h
The vertical component of the plane's velocity is given by:
y = 300 km/h * sin(300°) = 300 km/h * sin(60°) = 300 km/h * √3/2 = 150√3 km/h
The velocity of the wind relative to the ground is given as 85 km/h in the direction 225° from East. We can break this down into horizontal and vertical components using trigonometry.
The horizontal component of the wind's velocity is given by:
x = 85 km/h * cos(225°) = 85 km/h * (-√2/2) = -85√2/2 km/h
The vertical component of the wind's velocity is given by:
y = 85 km/h * sin(225°) = 85 km/h * ( -√2/2) = -85√2/2 km/h
To find the velocity of the plane relative to the air, we subtract the velocity of the wind from the velocity of the plane component-wise.
The horizontal component of the velocity of the plane relative to the air is given by:
150 km/h - (- 85√2/2 km/h) = 150 km/h + 85√2/2 km/h
The vertical component of the velocity of the plane relative to the air is given by:
150√3 km/h - (- 85√2/2 km/h) = 150√3 km/h + 85√2/2 km/h
Therefore, the velocity of the plane relative to the air is approximately:
150 km/h + 85√2/2 km/h in the horizontal direction,
150√3 km/h + 85√2/2 km/h in the vertical direction.