Kartik has three solid objects, a cone, a hemisphere, and a cylinder. All three have the same base radius and the same height. He completely immerses each solid in a bucket full of water. What is the ratio of the volumes of the cylinder: cone: hemisphere?

Let the common radius be $r$.

Since hemisphere has height equal to its radius, common height is also $r$.

Volume of cylinder:

$\pi r^2h=\pi r^2(r)=\pi r^3$

Volume of cone:

$\frac{1}{3}\pi r^2h=\frac{1}{3}\pi r^3$

Volume of hemisphere:

$\frac{2}{3}\pi r^3$

So, the ratio is:

$\pi r^3:\frac{1}{3}\pi r^3:\frac{2}{3}\pi r^3$

$=1:\frac{1}{3}:\frac{2}{3}$

Multiplying by $3$,

$=3:1:2$

Therefore, the correct answer is option $2$.