Answer
The girl's linear speed is $~~\frac{mvR^2}{I+MR^2}$
Work Step by Step
Note that the initial angular momentum of the system is zero.
We can use conservation of angular momentum to find the angular speed of the merry-go-round:
$L_f = L_i$
$(I+MR^2)~\omega - mvR = 0$
$(I+MR^2)~\omega = mvR$
$\omega = \frac{mvR}{I+MR^2}$
We can find the girl's linear speed:
$v_g = \omega~R$
$v_g = (\frac{mvR}{I+MR^2})~R$
$v_g = \frac{mvR^2}{I+MR^2}$
The girl's linear speed is $~~\frac{mvR^2}{I+MR^2}$
