Question

A body covered a distance of L metre along a curved path of a quarter circle. The ratio of distance to displacement.
(A) $\dfrac{\pi }{{2\sqrt 2 }}$
(B) $\dfrac{{2\sqrt 2 }}{\pi }$
(C) $\dfrac{\pi }{{\sqrt 2 }}$
(D) $\dfrac{{\sqrt 2 }}{\pi }$

Answer 1

Hint:In order to solve this question, we will first find the total distance covered by the body in terms of quarter circle length and radius and then we will find the displacement which is just the distance between initial and final point of the journey and therefore we will find the ratio of the distance to the displacement.



Complete answer:
Let us draw a quarter circle AOB right angled at O of radius R and AB represents the displacement of the body while the curved path of the quarter circle is the length L covered by the body as given in the question.

Now, the quarter circle has a perimeter of $\dfrac{C}{4}$ of total circle of radius R and as we know that perimeter of circle is $
  C = 2\pi R \\
   \Rightarrow \dfrac{C}{4} = \dfrac{{\pi R}}{2} \\
 $ and this length is equal to length covered by the body which is given to us as L so,
so, Distance is represented by L and it’s given by $L = \dfrac{{\pi R}}{2} \to (i)$
Now, as displacement is the distance covered by the body between initial and final point which A and B in the diagram which is AB, so OAB is a right angled triangle hence, using Pythagorean theorem we have,
${(AB)^2} = {(OA)^2} + {(OB)^2}$
on putting the values we get,
$
  {(AB)^2} = {(R)^2} + {(R)^2} \\
  AB = \sqrt 2 R \to (ii) \\
 $
as AB represent displacement covered by the body, so taking ratio of distance L from equation (i) to the displacement AB from equation (ii) we get,
$
  \dfrac{L}{{AB}} = \dfrac{{\dfrac{{\pi R}}{2}}}{{\sqrt 2 R}} \\
   \Rightarrow \dfrac{L}{{AB}} = \dfrac{\pi }{{2\sqrt 2 }} \\
 $
So, the ratio of distance to displacement is $\dfrac{\pi }{{2\sqrt 2 }}$

Hence, the correct option is Option (A).


Note:While solving such questions, don’t confuse distance and displacement as the same quantity because distance is just total length covered by the body and displacement is actual length between initial and final point of the journey.