Final answer:
To find the plane's speed relative to the ground, we add the vector of the plane's speed through the air to the wind's velocity vector, then calculate the magnitude and direction of the resultant vector using vector addition and trigonometry.
Step-by-step explanation:
To calculate the speed of the plane relative to the ground, we must first determine the velocity vectors of the plane's speed relative to the air and the wind’s velocity. Then we add the two vectors to find the resultant vector, which is the plane's velocity relative to the ground.
Let the vector representing the plane's velocity relative to air be A, given as 175 km/h [E 49º S]. The vector representing the wind’s velocity, B, is given as 40.0 km/h [E 35º N]. To find the resultant velocity (vector R), we can use vector addition:
A = 175 km/h at 49º south of east
B = 40 km/h at 35º north of east
R = A + B
We break down vectors A and B into their horizontal (East-West) and vertical (North-South) components. Summing the horizontal and vertical components separately gives us the components of R, from which we can then compute the magnitude and direction of vector R using trigonometry.