Question

what is relation between magnification of spherical mirror and velocity of image

Answer 1

Convex mirror :

R =2 m .    So  f = R/2 = 1 m.

Mirror equation:

    1/v + 1/u = 1/f.
    Here f is +ve, u is -ve. v is positive.

To know the speeds of the image, differentiate wrt time.

    -1/v² * dv/dt - 1/u² * du/dt = 0.            f is constant.
     dv/dt = - v²/u² * du/dt

=>  dv/dt = - m²  * du/dt

Given  du/dt = 5 m/s   
  (u is negative. du/du is +ve means u is increasing as runner approaches
    the mirror.  It is true as u is -ve.)

     v = u f / (u - f)
     v²/u² = m² = f²/ (u - f)²

=>  dv/dt = - 5 f²/(u-f)²

a)  f =  1m.   u = -39m.
     dv/dt = - 5 * 1²/40²   m/s

b)  f = 1m,  u = -29m
       dv/dt = - 5 * 1²/30²  m/s

c)  f = 1m.  u = -19m
       dv/dt = - 5 * 1²/20²  m/s

d) f = 1m.  u = - 9m
      dv/dt = - 5 * 1² /10²  m/s