To find the ground speed and direction of the plane, we need to break down the velocity of the plane and the wind velocity into their horizontal and vertical components.
Given:
Airspeed of the plane = 724 km/h
Bearing of the plane = North 30 degrees East
Wind velocity = 32 km/h from the west (or directly east)
Step 1: Find the horizontal and vertical components of the plane's velocity:
To do this, we can use trigonometry.
Horizontal component = Airspeed * cos(Bearing)
Vertical component = Airspeed * sin(Bearing)
Horizontal component = 724 * cos(30°)
Vertical component = 724 * sin(30°)
Step 2: Find the horizontal and vertical components of the wind velocity:
In this case, since the wind is blowing directly east (from the west), the horizontal component of the wind velocity will be 32 km/h, and the vertical component will be 0 km/h.
Horizontal component of wind velocity = 32 km/h
Vertical component of wind velocity = 0 km/h
Step 3: Add up the horizontal and vertical components of the plane and wind velocity separately:
Horizontal component of total velocity = Horizontal component of plane's velocity + Horizontal component of wind velocity
Vertical component of total velocity = Vertical component of plane's velocity + Vertical component of wind velocity
Step 4: Calculate the magnitude and direction of the total velocity:
Magnitude of the total velocity (ground speed) = √(Horizontal component of total velocity)^2 + (Vertical component of total velocity)^2
Direction of the total velocity = tan^(-1)(Vertical component of total velocity / Horizontal component of total velocity)
Step 5: Substitute the values into the formulas and solve:
Horizontal component of total velocity = 724 * cos(30°) + 32
Vertical component of total velocity = 724 * sin(30°) + 0
Magnitude of the total velocity = √((724 * cos(30°) + 32)^2 + (724 * sin(30°))^2)
Direction of the total velocity = tan^(-1)( (724 * sin(30°)) / (724 * cos(30°) + 32) )
Calculating these values will give you the ground speed of 740.5 km/h and the direction of 57.9 degrees, as provided by the teacher's solution.