Answer
$h=\frac{7}{5}r$
Work Step by Step
We can determine the required height as follows:
According to the impulse and angular momentum principle
$H_{A_1}+r\int Fdt=H_{A_2}$
$\implies I_A\omega_1+h\int Pdt=I_A\omega_2$
$\implies (\frac{2}{5}mr^2+mr^2)(0)+P\Delta th=()\omega_2$
This simplifies to:
$P\Delta th=(\frac{2}{5}mr^2+mr^2)\omega_2$ [eq(1)]
We know that
$mv_1+\int Fdt=mv_2$
$\implies m(0)+P\Delta t=mv_2$
$\implies P\Delta t=mr\omega_2$ [eq(2)]
We plug in this value from eq(2) into eq(1) to obtain:
$mr\omega_2h=(\frac{2}{5}mr^2+mr^2)\omega_2$
This simplifies to:
$h=\frac{7}{5}r$