Question

An aeroplane has to go from a point $A$ to another point $B500km$ away due ${30}^0$ east of north. A wind is blowing due north at a speed of $20m/s$. The air-speed of the plane is $150m/s$.
(a) Find the direction in which the pilot should head the plane to reach the point $B$.
(b) Find the time taken by the plane to go from $A$ and $B$.

Answer 1

In resultant direction $R$ the plane reach the point $B$

Velocity of wind ${\overrightarrow{V}}_W=20m/s$

Velocity of aeroplane ${\overrightarrow{V}}_a=150m/s$

In $△ACD$ according to sine formula

$\therefore \frac{20}{\sin A}-\frac{150}{\sin {30}^o}\Rightarrow \sin A-\frac{20}{150}\times \frac{1}{2}-\frac{1}{15}$

$\Rightarrow A={\sin }^{-1}\left(1/15\right)$

a) The direction is ${\sin }^{-1}\left(1/15\right)$ east to the line $AB$

b) ${\sin }^{-1}\left(1/15\right)=3^o{48}^{\prime }$

$\Rightarrow {30}^o+3^o{48}^{\prime }={33}^o{48}^{\prime }$

$R=\sqrt{{150}^2+{20}^2+2\left(150\right)20\cos {33}^o{48}^o}=167m/s$

Time $=\frac{s}{v}-\frac{500000}{167}=2994sec=49=50min$