As shown in figure `AB` represents an infinite wall tangential to a horizontal semi circular track.`O` is a point source of light on the ground at the centre of the circle. A block moves along the circular track with speed `v` starting from the point where the wall touches the circle. If the velocity and acceleration of shadow along the length of the wall is respectively `'V'` and `'a'` then,

Options

A. `V = v cos (vt)/(R)`
B. `V = v sec^(2) ((vt)/(R))`
C. `a = (v^(2))/(R) sec^(2) ((vt)/(R)) tan ((vt)/(R))`
D. `a = (2v^(2))/(R) sec^(2) ((vt)/(R)) tan ((vt)/(R))`

Answer

B, D

Explanation

`tan theta = (y)/(R) , y = R tan theta `
`v = (dy)/(dt) = R sec^(2)theta ((d theta )/(dt)) = R omega^(2) sec^(2) theta `
`= v sec^(2) ((vt)/(R))`
Acceleration of the shadow :
`a = (dv)/(dt) = (2v^(2))/(R) sec ^(2) ((vt)/(R)) tan ((vt)/(R))`