Interpret the setup

• A wheel of mass $M=12\,\text{kg}$ rolls down the incline $PQ$ without slipping.
• It is connected by a light string over a pulley to a hanging/block mass $m=3\,\text{kg}$.
• The string has fixed length, so both bodies have the same speed magnitude $v$.
• Surface $PR$ is frictionless, so after reaching $Q$, the wheel moves on the horizontal without rotational constraint from friction. But the question asks the speed when it reaches $Q$, so we only need the motion up to $Q$.

Use energy conservation

Let the vertical drop of the wheel's centre from $P$ to $Q$ be $h$.

From the geometry shown in such standard problems, when the wheel goes from $P$ to $Q$, the hanging mass rises by the same amount $h$ because the string length is fixed.

Hence:

• Loss in gravitational potential energy of wheel = $12gh$
• Gain in gravitational potential energy of $3\,\text{kg}$ mass = $3gh$

So net decrease in potential energy is $12gh-3gh=9gh$

Write final kinetic energy

At $Q$, both have speed $v$.

• Translational KE of wheel: $\frac12 Mv^2=\frac12(12)v^2=6v^2$

• Rotational KE of rolling wheel:

  For a wheel (ring), $I=MR^2$ and since rolling without slipping, $\omega=\frac{v}{R}$ Therefore

  =\frac12(MR^2)\left(\frac{v}{R}\right)^2 =\frac12 Mv^2=6v^2$$

• KE of $3\,\text{kg}$ mass: $\frac12 mv^2=\frac12(3)v^2=\frac32 v^2$

Total kinetic energy: $6v^2+6v^2+\frac32 v^2=\frac{27}{2}v^2$

Apply conservation of energy

$9gh=\frac{27}{2}v^2$

So, $v^2=\frac{18gh}{27}=\frac{2gh}{3}$

Hence, $v=\sqrt{\frac{2gh}{3}}$

Match with given form

Given, $v=\frac12\sqrt{xgh}$

Therefore, $\frac14 xgh=\frac{2gh}{3}$

Cancelling $gh$, $\frac{x}{4}=\frac{2}{3}$ $x=\frac{8}{3}$

Check against stored answer

My derived value is $x=\frac{8}{3}$ which is not equal to the stored answer $3$.

So I disagree with the stored answer.

The only way to get an integer near this is if a different moment of inertia were intended. For example, for a solid disc $I=\frac12 MR^2$, one gets a different value, but still not $3$. Thus the stored answer likely does not match the standard interpretation of a "wheel" as a ring/hoop in rolling motion with the shown displacement relation.