Question

A body moves along the curved path of quarter circle . Calculate the ratio of distance to displacement​

Answer 1

DATA GIVEN –

Body moves along the curved path of only quarter of a circle.

TO FIND –

Ratio between distance and displacement.

ANSWER AND EXPLANATION –

Say the radius of the circle be ' r '.

Distance indicates the total length of the path covered by the person. Since the person covers only the curved part of a quarter of a circle, the distance = πr/2

And to answer how,

circumference of a circle = 2πr

Since the distance covered by the person is only one fourth of the circumference of a circle. We can say that the distance covered by the person = 2πr/4 = πr/2

So the distance ( d ) = πr/2 .

Now coming to the point of displacement, it is nothing but the length of the shortest path possible between initial and final points.

In the case of a quadrant, the term displacement indicates the hypotenuse of the two radii which are taken as sides.

We know that,

$\\ hypotenuse {}^{2} \: = side {}^{2} + side {}^{2}$

Here the hypotenuse indicates the displacement ( s ) and the sides indicate the radii ( r ).

So,

$s {}^{2} = r { }^{2} + r {}^{2}$

$s {}^{2} = 2r {}^{2}$

$s = \sqrt{2r {}^{2} } = \sqrt{2} \: r$

So displacement ( s ) =

$\sqrt{2} \: r$

Ratio of Distance and Displacement

= d : s = d/s

$= \: \frac{ \frac{\pi \: r}{2} }{ \sqrt{2} \: r } = \frac{\pi \: r}{2 \sqrt{2} \: r} = \frac{\pi}{2 \sqrt{2} }$

So, π: (2√2) is the answer.

In the terms of numbers, the answer = 11 : 7√2 since π = 22/7.

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Answer 2

Step 1: Given data

Body moves along the curved path of quarter circle.

radius of circle$=r$

$\frac{distance}{displacement}=\frac{d}{D} =?$

Step 2: Using the formula

circumference of a circle$=2\pi r$

$hypotenuse^{2}=base^{2} + height^{2}$

Step 3: Calculating the required ratio

The distance covered by the person is only one fourth of the circumference of a circle.

Distance covered is calculated as,

$d=\frac{2\pi r}{4} =\frac{\pi r}{2}$

The displacement is the length of the shortest path between the starting and ending points.

In the case of a quarter of a circle, the displacement represents the hypotenuse and the two radii used as sides.

$D^{2}=r^{2} +r^{2} =2r^{2}$

$D =\sqrt{2} r$

Thus, required ratio,

$=\frac{d}{D} =\frac{\frac{\pi r}{2} }{\sqrt{2}r }=\frac{\pi }{2\sqrt{2} }$

put $\pi =\frac{22}{7}$

$\frac{d}{D} =\frac{\frac{22}{7} }{2\sqrt{2} } =\frac{11}{7\sqrt{2} }$

Hence, the ratio of distance to displacement is $11:7\sqrt{2}$.

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