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 $2$
$l_2=\pi r=\pi \left(AC\right)=\pi Rsin\frac{\pi }{3}$
$=\frac{\sqrt{3}}{2}\pi R=0.866\pi R$
$\therefore l_1<l_2$