Question

An athlete completes one round of a circular track of diameter 200 m in 40 seconds. What will be the distance covered and displacement at the end of 2 minutes 20 seconds?

Answer 1

We are given the time taken by an athlete to complete one round on a circular path and the diameter of the path is also given. To find the distance traveled by the athlete in the given time, first we need to find the total number of rounds he completed in the given time. Then by multiplying it with the perimeter of the path we will get the total distance. While calculating the total rounds and distance we will find the final position of the athlete. Thus we can also find the displacement of the athlete in the given time.
Formula used:
Perimeter or circumference of a circle, $C=2\pi r$

Complete answer:
In the question it is said that an athlete completes one round of a circular track in 40 seconds.
The diameter of the rack is given to be 200 m.

Consider the figure given below.

In the figure AB is the diameter, $d=200m$
Therefore radius of the circular track,
$\begin{aligned} & r=\dfrac{d}{2} \\\ & \Rightarrow r=\dfrac{200}{2}=100m \\\ \end{aligned}$
As in the given situation, the athlete starts from position A and reaches A in 40 seconds.
Time taken to complete one round, ${{t}_{0}}=40\sec$
We need to find the distance and displacement traveled by the athlete in 2 minutes 20 seconds, i.e.
$t=2\min 20\sec$
Let us convert this into seconds.
$\Rightarrow t=\left( 2\times 60 \right)+20\sec$
$\Rightarrow t=120+20=140\sec$
Now we can find the number of rounds completed by the athlete in time ‘t’.
This will be,
$n=\dfrac{t}{{{t}_{0}}}$, were ‘n’ is the number of rounds completed in time ‘t’.
$\Rightarrow n=\dfrac{140}{40}$
$\Rightarrow n=3.5$
Therefore the number of rounds completed is 3.5, which means the athlete runs 3 times the full track and one time half of the track.

By considering the figure we can say that, the athlete runs 3 times from point A to A and last time he runs from point A to B (the semicircular path from A to B is half of the circular path).
We know that the product of the number of rounds completed by the athlete and perimeter of the track will give the total distance covered by him.
Perimeter or circumference of the circular path can be given as,
$C=2\pi r$
Therefore we get the total distance covered,
$\text{distance}=n\times 2\pi r$
By substituting the known values in the above equation we get
$\Rightarrow \text{distance}=3.5\times 2\times \dfrac{22}{7}\times 100$
$\Rightarrow \text{distance}=2200m$
Hence the total distance covered by the athlete is 2200 m.
Now we need to find the displacement of the athlete.
Considering the figure, we know that starting from point A after 3.5 rounds the final position of the athlete is at point B.

We know that the displacement is the minimum distance from starting point to final point.
Since here the starting point is ‘A’ and the ending point is ‘B’, the minimum distance between them is the diameter of the circular path.
Therefore the displacement of the athlete is 200 m.
Thus we have the distance and displacement covered by the athlete in 2 minutes 20 seconds as 2200 m and 200 m respectively.

Note:
Distance is a scalar quantity. We know that scalar quantity is a physical quantity that has only magnitude. Hence distance is the total movement of a body regardless of direction. It can also be defined as the measure of total ground covered by an object during its motion.
Distance can be positive and zero. Distance of a body can never be a negative value.
Displacement is a vector quantity, i.e. it has both magnitude and direction. It is simply defined as the change in position of a body. Displacement of a body can be negative, positive and zero.