A block is projected with a speed `v_0` such that it strikes the point of projection after describing the path as shown by the dotted line. If friction exists for the parth of length `d` and the vertical circular path is smooth, assuming `mu`=coefficient of friction,

a. Find `v_0`.
b. What is the minimum value of `v_0`?

Answer

a. `sqrt((d^3g)/(4R)+2g[mud+2R])`; b. `sqrt(5gR)`

Explanation

a. From C to A, time taken
`t=sqrt((2[2R])/(g))=sqrt((4R)/(g))`
`v_2=d/t=dsqrt((g)/(4R))`
From B to C, `1/2mv_1^2=1/2mv_2^2+mg(2R)`
`impliesv_1^2=v_2^2+4gR=(d^2g)/(4R)+4gR`
From A to B: `v_1^2=v_0^2-2mugdimpliesv_0^2=v_1^2+2mugd`
`impliesv_0=sqrt((d^2g)/(4R)+2g[mud+2R])`
b. For minimum `v_0`, `v_1=sqrt(5gR)`