An athlete runs on a circular track of radius r meters with a uniform speed and completes one revolution in t seconds. Calculate the distance and displacement in:
1. t/2 s
2. 3t/4 s
Answers
kindly refer to the attachment for the answer
Concept:
The displacement is a vector quantity that has magnitude as well as directions.
Given:
Athlete runs on a track of radius r and the time for one revolution is t.
Find:
The distance and displacement at t/2 s and 3t/4 s.
Solution:
In the figure attached, O is the initial point, A is after time t/2 and B is after 3t/4.
Distance traveled is the actual length traveled.
As an athlete requires t time for 1 revolution, 1 revolution is equal to 2πr.
In t seconds, 2πr distance is traveled.
In 1 second, 2πr/t distance is traveled.
So, in t/2 seconds, the distance traveled is (t/2)×(2πr/t)= πr m.
Similarly, in 3t/4 seconds,
The distance traveled is (3t/4)×(2πr/t)= 3πr/2 m.
Displacement is the shortest distance between the initial and final points.
In t/2 seconds, it will be at point A.
So, the shortest distance will be 2r m.
Similarly, in 3t/4 seconds, it will be at point B.
The displacement will be
D = √[r²+r²] = √[2r²] = √2r m
So, the shortest distance will be √2r m.
Hence, at t/2 seconds, distance is πr m and displacement is 2r m and at 3t/4 seconds, distance is 3πr/2 m and displacement is √2r m.
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