A particle is moving along a circle with a uniform speed `v`. Find `(a)` change in the magnitude of velocity and `(b)` the magnitude of change in the velocity when it has rotated an angle `theta . (0 lt theta lt 90^(@))`

Explanation

(a) Since the particle is moving with a uniform speed , the magnitude of velocity `|vec (v)| = v` , remains same , hence the change in the magnitude of velocity , i.e. `Delta | vec (v)| = 0`.
(b) The magnitude of change in the velocity `|Delta vec(v)|`:
OR
We can represent
`Delta v = vec(v_(2)) - vec(v_(1)) = vec(v_(2)) + (- vec(v_(1)))`
`Delta v = sqrt(v^(2) + v^(2) + 2v . v cos (180 - theta))`
`= sqrt( 2v^(2) - 2v^(2) cos theta)` `[since cos (180 - theta) = - cos theta]`
` = sqrt(2) v sqrt( 1 - cos theta)`
` = sqrt(2) v sqrt(( 1 - ( 1 - 2 sin^(2) (theta)/(2))`

`= 2v sin ((theta)/(2))`
Change in the velocity ` = | Delta vec (v)| = Delta v = 2v sin ((theta)/(2))`.
If `theta = 90^(@) , Delta v = sqrt(2) v` (as in the previous problem).