Question

A glider aircraft initially traveling due west at 87.0 km/h encounters a sudden gust of wind at 39.5 km/h directed toward the northeast (see the figure below). What are the speed and direction of the glider relative to the ground during the wind gust? (The velocity of the glider with respect to the ground is the velocity of the gilder with respect to the wind plus the velocity of the wind with respect to the ground.)

Answer 1

Let the four directions be such that

North = +y

South = -y

East = +x

West = -x

Initial velocity of the glider = 87.0 km/h towards west.

We can write this as: v_i = -87 i km/h

Velocity of wind v' = 39.5 km/h towards North East.

(Assuming since no plot is given, the direction is exactly north east which means 45 degrees from east (or x axis)) Thus, we can write wind speed as:

v'= 39.5 cos (45) + 39.5 sin(45) km/h = 27.93 i + 27.93 j km/h

Now the net speed of glider with respect to earth, after it encounters the wind:

v_f = v_i + v'

v_f = -87 i + (27.93 i + 27.93 j ) = ( -87 + 27.93) i + 27.93 j

v_f = -59.1 i + 27.93 j km/h

Magnitude of final velocity:

v_f = square root ((-59.1)² + 27.93²)

v_f = square root (3491.04+780.2) = square root (4271.24) = 65.355 km/h

Thus magnitude of v_f = 65.36 km/h

Direction of v_f:

tan (theta) = -27.93/59.1 = -0.473

or, Since tan (-theta) = -tan (theta)

We have, theta = tan-1 (-0.473) = -25.3 degrees or direction is 25.3 degrees north of west.

(25.3 degrees from