An airplane has an airspeed of 724 kilometers per hour at a bearing of 30 degrees.The wind velocity is 32 kilometers per hour from the west. What are the groundspeed and the direction of the plane?
To determine the groundspeed and direction of the plane, we need to consider the effect of the wind on the plane's motion.
Step 1: Resolve the wind velocity into its north-south (crosswind) and east-west (headwind/tailwind) components.
Given:
Wind velocity = 32 kilometers per hour (from the west)
Since the wind is coming from the west (270 degrees), its east-west component is -32 km/h (negative because it opposes the plane's motion in the east direction) and the north-south component is 0 km/h.
Step 2: Calculate the net velocity of the plane in the east-west and north-south directions.
To do this, we can use vector addition. We add the velocities given by the airspeed and the wind velocity components.
East-West Component:
Airplane airspeed: 724 km/hr * cos(30) = 724 * √(3)/2 ≈ 626.12 km/hr (positive because it's in the east direction)
Wind component: -32 km/hr (negative because it opposes the airplane's eastward motion)
Net east-west velocity = 626.12 km/hr - 32 km/hr = 594.12 km/hr
North-South Component:
Airplane airspeed: 724 km/hr * sin(30) = 724 * 1/2 = 362 km/hr (positive because it's in the north direction)
Wind component: 0 km/hr
Net north-south velocity = 362 km/hr + 0 km/hr = 362 km/hr
Step 3: Calculate the groundspeed and direction.
Using the net east-west and north-south velocities, we can apply the Pythagorean theorem to find the groundspeed.
Groundspeed = √(net east-west velocity^2 + net north-south velocity^2)
Groundspeed = √(594.12^2 + 362^2)
Groundspeed ≈ √(352985.6544 + 131044) ≈ √484029.6544 ≈ 695.284 km/hr
Lastly, we need to determine the direction (bearing) of the plane relative to the north.
Direction = arctan(net east-west velocity / net north-south velocity)
Direction = arctan(594.12 km/hr / 362 km/hr)
Direction ≈ arctan(1.6409) ≈ 57.213 degrees
Therefore, the groundspeed of the plane is approximately 695.284 km/hr, and its direction is approximately 57.213 degrees relative to the north.