Question

An aeroplane is flying from city $ A $ to city $ B $ along path $ 1 $ . The path $ 1 $ is a circular arc whose centre coincides with the centre of the earth. Another aeroplane is flying along path $ 2 $ from $ A $ to $ B $ . The path $ 2 $ is circular arc whose centre is at $ C $ . $ O $ is the centre of the earth. Then,

• A. The distance travelled by $ 1^{st} $ aeroplane is greater than that of $ 2^{nd} $ aeroplane
• B. The distance travelled by $ 1^{st} $ aeroplane is less than that of $ 2^{nd} $ aeroplane
• C. The displacement of both aeroplane is different
• D. None of these

Answer 1

Answer: (B)

Let $ l_1 = $ distance travelled by aeroplane $ 1 $ along path $ 1 $
$ l_2 = $ distance travelled by aeroplane $ 2 $ along path $ 2 $
Let $ OC=\frac{R}{2} $

Here, for path $ 1 $ , $ 2\theta=\frac{l_{1}}{R} $
But $ cos\,\theta=\frac{R}{2R}=\frac{1}{2} $
$ \therefore \theta=\frac{\pi}{3} $
$ \Rightarrow 2 \times\frac{\pi}{3}=\frac{l_{1}}{R} $
$ \therefore l_{1}=\frac{2\pi R}{3}=0.67\,\pi R $
But for path $ 2 $
$ l_{2}=\pi r=\pi\left(AC\right)=\pi R\,sin \frac{\pi}{3} $
$ =\frac{\sqrt{3}}{2} \pi R=0.866\,\pi R $
$ \therefore l_{1} < l_{2} $