Constraint motion of Blocks!

Call the speed of the hanging mass $V_h$ and the speed of the mass on the ground $V_g$ .

As you can see from the picture, when the hanging mass moves down by a distance $dh$ the rope gets shorter by an amount $2dh$ . This extra length can only come from the length $L$ getting shorter. Therefore:

$|\frac{dL}{dt}|=2|\frac{dh}{dt}|$ .

You can see that the component of $\vec V_g$ in the direction of $\hat L$ is:

$\vec V_g\cdot \hat L=V_g\cos(\theta)=|\frac{dL}{dt}|$

It is also obvious that $V_h=\frac{dh}{dt}$ .

Therefore we have the following relationship between $V_g$ and $V_h$

$V_g=\frac{2}{\cos(\theta)}V_h$

Now we can apply conservation of energy:

$mgh=0.5(mV_h^2+3mV_g^2)=0.5mV_h^2(1+3(\frac{2}{\cos(\theta)})^2)\Rightarrow V_h=\sqrt{\frac{2gh}{1+12\sec(\theta)^2}}$

Now just plug in $\theta=60°$ and you will get $V_h=\frac{\sqrt{2gh}}{7}$