Answer to Question #349053 in Analytic Geometry for Mapula Advice

Question

Question text

An aeroplane flies at a ground velocity (i.e. velocity relative to

the ground) of 300 km/h N 30o W, in a wind blowing at a velocity

of 50 km/h N 20o E. What is the velocity (speed and direction)

of the plane relative to the ground? (Use a calculator and round the speed

to the nearest km/h, and the corresponding angle to the nearest degree.)

Answer 1

a.

$$\vec{v}=300\cos120\degree \vec{i}+300\sin120\degree\vec{j}$$

$$\vec{u}=50\cos70\degree \vec{i}+50\sin70\degree\vec{j}$$

$$\vec{v}_{res}=\vec{v}+\vec{u}$$

$$=(300\cos120\degree+50\cos70\degree )\vec{i}$$

$$+(300\sin120\degree+50\sin70\degree)\vec{j}$$

$$\approx-132.9\vec{i}+306.8\vec{j}$$

$$|\vec{v}_{res}|\approx\sqrt{(-132.9)^2+(306.8)^2}\approx334.35(km/h)$$

b.

$$\tan \theta=\dfrac{306.8}{-132.9}=-2.3085$$

$$\theta=\tan^{-1}(-2.3085)+180\degree\approx113.42\degree$$

An aeroplane travels in the direction N $23.42\degree$W at $334.35$ km/h.