Question

A body is moving along the circumference of a circle of radius 'R' and completes half of the revolution. Then, the ratio of its displacement to distance is
A. \[\pi :2\]
B. \[2:1\]
C. \[2:\pi \]
D. \[1:2\]

Answer 1

Hint: We will use the basic understanding of distance and displacement of the given body. We know that the diameter of a circle is twice the radius of that circle. Using these concepts, we will deduce the final expression for the ratio of displacement and distance of the given body.

Complete step by step answer:
One complete revolution of a circle is equal to the circle's circumference, and it is given that the distance travelled by the body is half of its complete revolution. This means that the given body's distance is equal to half of the circumference of the circle. We can express the circumference of the circle as below:
\[C = 2\pi R\]
Using the above explanation, we can write the displacement of the circle as below:
\[d = \dfrac{1}{2}C\]
Here d is the distance covered by the body while moving on the circumference of the circle.
On substituting \[2\pi R\] for C in the above expression, we get:
\[\begin{array}{c}
d = \dfrac{1}{2}\left( {2\pi R} \right)\\
 = \pi R
\end{array}\]……(1)
We know that the displacement of a body is the minimum distance traveled by it. As the body completes half of the revolution, we can say that displacement covering its diameter gives the body. We also know the relationship between diameter and radius of the circle can be expressed as:
\[D = 2R\]
Here D is the diameter of the circle.
Based on the above explanation, we can write the expression for displacement covered by the body as below:
\[d' = 2R\]……(2)
Here \[d'\] is the displacement of the given body.
On dividing equation (1) and equation (2), we get:
\[\begin{array}{c}
\dfrac{d}{{d'}} = \dfrac{{\pi R}}{{2R}}\\
 = \pi :2
\end{array}\]
Therefore, we can say that the ratio of displacement to distance is \[\pi :2\]

So, the correct answer is “Option A”.

Note:
It would be an added advantage if we remember that a circle's diameter is twice of its radius. We can also remember the expression for the circumference of the given circle that is directly proportional to its radius.