an aeroplane flys with an air speed of 40m/h at a bearing of 20degrees while at the same time is blowing at 100km/h at a bearing of 90degrees.now determine the speed and direction of the plane as seen by the observer on the ground
To determine the speed and direction of the plane as seen by the observer on the ground, we can use vector addition.
Step 1: Convert the air speed of the plane and the wind speed to the same unit.
The air speed of the plane is given as 40 m/h (miles per hour), and the wind speed is given as 100 km/h (kilometers per hour). We need to convert these to the same unit. Let's convert everything to kilometers per hour.
1 mile = 1.60934 kilometers
1 hour = 60 minutes (for consistent units)
So, the air speed of the plane is 40 m/h * 1.60934 km/m ≈ 64.37 km/h.
Step 2: Break down the air speed and wind speed into their respective horizontal and vertical components.
The air speed of the plane has a bearing of 20 degrees. To break it down into horizontal and vertical components:
Horizontal component = 64.37 km/h * cos(20)
Vertical component = 64.37 km/h * sin(20)
The wind speed has a bearing of 90 degrees, which means it is blowing directly towards the north. So, the horizontal component of wind speed is 0 km/h, and the vertical component is 100 km/h.
Step 3: Add the horizontal and vertical components of the plane's air speed and wind speed.
Horizontal component of the plane's speed = Horizontal component of air speed + Horizontal component of wind speed
Vertical component of the plane's speed = Vertical component of air speed + Vertical component of wind speed
Horizontal component of the plane's speed = 64.37 km/h * cos(20) + 0 km/h = 64.37 km/h * cos(20)
Vertical component of the plane's speed = 64.37 km/h * sin(20) + 100 km/h = 64.37 km/h * sin(20) + 100 km/h
Step 4: Use the horizontal and vertical components of the plane's speed to find the magnitude and direction.
Magnitude of the plane's speed = √((Horizontal component)^2 + (Vertical component)^2)
Direction of the plane's speed = arctan(Vertical component / Horizontal component)
Magnitude of the plane's speed = √((64.37 km/h * cos(20))^2 + (64.37 km/h * sin(20) + 100 km/h)^2)
Direction of the plane's speed = arctan((64.37 km/h * sin(20) + 100 km/h) / (64.37 km/h * cos(20)))
By plugging in the values and evaluating these equations, you will get the magnitude and direction of the plane's speed as seen by the observer on the ground.