A particle starts moving on a circular track of radius `(20)/(pi) m` from point A with a uniform speed of `20 m//s`. It reaches to point B after some time. Find :
(a) Distance travelled by particle from A to B and its displaement for the same duration.
(b) Find the average speed, average velocity and average acceleration from A to B

Explanation

(a) Length of arc `AB = (2 pi R)/(4) = (pi R)/(2) = (pi)/(2) xx (20)/(pi) = 10 m`
So, distance travelled from A to B is 10 m. Taking 'O' as origin,
initial position vector of particle, `vec(r)_(i) = vec(OA) = (20)/(pi) hat(i)m`
final position vector of particle, `vec(r)_(f) = vec(OB) = (20)/(pi) hat(j)m`
`:.` Displacement `= Delta vec(r) = vec(r)_(f) - vec(r)_(i) = (20)/(pi) (hat(j) - hat(i))m`
(b) Time taken by particle to reach from A to B.
`t = ("distance")/("speed") = (10)/(20) = 0.5 s`
Average speed `= ("distance")/("time") = (10)/(0.5) = 20 m//s`
Average velocity `= ("displacement")/("time") = (Delta vec(r))/(Delta t) = ((20)/(pi) (hat(j) - hat(i)))/(0.5)`
`= (40)/(pi) (hat(j) - hat(i)) m//s`
Average acceleration `= ("Change in velocity")/("Time")`
`= (vec(v)_(f) - vec(v)_(i))/(Delta t) = (-20 hat(i) - (20 hat(j)))/(0.5)`
`= - 40 (hat(i) + hat(j)) m//s^(2)`