Question

An object is moving in a circle of the radius $100m$ with a constant speed of $31.4m/s$. What is its average speed for one complete rotation?
A.$0$
B.$31.4m/s$
C.$3.14m/s$
D.$\sqrt 2 \times 31.4m/s$

Answer 1

Hint: To reach the answer, we must carefully analyze the distance covered by the object depending on the shape of the path followed and the time taken by the object to cover this distance. We should also focus on the definition of the average speed and put all the determined values given in the question into the formula of the average speed to get the desired result.

Complete answer:
The ratio of the total distance traveled by the object and the total time taken by the object to cover this distance is known as the average speed of an object. Mathematically it is given by:
${v_{av}} = \dfrac{d}{T}.......(1)$
Where,
${v_{av}}$ is the average speed.
$d$ is the distance traveled.
$T$ is the time taken.
Now in the question, it is given that the object moves in a circular path and we know that the distance covered by the object in one complete rotation will be equal to the circumference of the circular path. Therefore,
$d = 2\pi r......(2)$
Where,
$d$ is the distance traveled.
$r$ is the radius of the circular path.
It is given that the radius,$r$ is $100m$. Substituting the value of radius in equation (2) we get,
$d = 2\pi \times 100$
$ \Rightarrow d = 200\pi \,m.......(3)$
Now, we have to find the time taken to travel this distance. We know that,
$speed = \dfrac{{dis\tan ce}}{{time}}......(4)$
In the question, the speed is given as $31.4m/s$. Substituting the value of speed and distance in equation (4) we get,
$31.4 = \dfrac{{200\pi }}{T}$
$ \Rightarrow T = \dfrac{{200\pi }}{{31.4}}\sec ......(5)$
We have to put the value of distance and time from equations (3) and (5) into equation (4) to get the average speed,
${v_{av}} = \dfrac{{200\pi }}{{\left( {\dfrac{{200\pi }}{{31.4}}} \right)}}$
$ \Rightarrow {v_{av}} = 31.4m/s$

So, option B is the correct answer.

Note:
The average speed and the average velocity are two different quantities. The average speed is a scalar quantity i.e. it only denotes the magnitude whereas the average velocity is a vector quantity representing both magnitude and direction. Also, the average velocity is defined as the net displacement covered per unit time and average speed is the total distance traveled per unit time.