Question

The compass of an airplane indicates that it is headed due north and its airspeed indicator shows that it is moving through the air at 240 km/hr. If there is a wind of 100 km/hr from west to east. (Hint: in this problem there is no need to convert to m/s)
a) Draw an appropriate diagram and determine the velocity of the plane relative to the earth?
b) What direction should the pilot head to travel due north and what then will be her velocity relative to the earth?

Answer 1

Answer:

Airplane speed relative to the ground is 260 km/h and θ = 22.6º  direction from north to east

Explanation:

This is a problem of vector composition, a very practical method is to decompose the vectors with respect to an xy reference system, perform the sum of each component and then with the Pythagorean theorem and trigonometry find the result.

Let's take the north direction with the Y axis and the east direction as the X axis

         Vy = 240 km / h            airplane

         Vx = 100 Km / h              wind

a) See the annex

Analytical calculation of the magnitude of the speed and direction of the aircraft

         V² = Vx² + Vy²

         V = √ (240² + 100²)

         V = 260 km/h

Airplane speed relative to the ground is 260 km/h

         Tan θ = Vy / Vx

         tan θ = 100/240

         θ = 22.6º

           

Direction from north to eastb

b) What direction should the pilot have so that the resulting northbound

          Vo = 240 km/h      airplane

          Vox = Vo cos θ

          Voy = Vo sin  θ

          Vx = 100 km / h      wind

To travel north the speeds the x axis (East) must add zero

         Vx -Vox = 0

         Vx = Vox = Vo cos θ

         100 = 240 cos θ

          θ = cos⁻¹ (100/240)

          θ = 65.7º

North to West Direction

The speed in that case would be

           V² = Vx² + Vy²

To go north we must find Vy

          Vy² = V² - Vx²

          Vy = √( 240² - 100²)

          Vy = 218.2 km / h