you have one degree on freedom which is $~S~$

from the constraint equation

$$ h^2+\left(\frac S2\right)^2=\left(\frac l2\right)^2\tag 1$$

you obtain

$$h=h(S)=\frac 12\sqrt{l^2-S^2}$$

thus $$\dot h=\frac{\partial h}{\partial S}\,\dot S= -\frac 12\,{\frac {S}{\sqrt {-{S}^{2}+{l}^{2}}}}\,\dot S$$

with $$\dot S=f(t)\quad ,S=\int f(t)\,dt+S_0$$

and $~S_0=\frac l2\,\cos(\theta)~$

you can obtain the velocity $~\dot h(t)~$ and the position $~h(t)=\int \dot h\,dt$

you have one degree on freedom which is $~S~$

from the constraint equation

$$ h^2+\left(\frac S2\right)^2=\left(\frac l2\right)^2\tag 1$$

you obtain

$$h=h(S)=\frac 12\sqrt{l^2-S^2}$$

thus $$\dot h=\frac{\partial h}{\partial S}\,\dot S= -\frac 12\,{\frac {S}{\sqrt {{l}^{2}-{S}^{2}}}}\,\dot S$$

with $$\dot S=f(t)\quad ,S(t)=\int f(t)\,dt+S_0$$

and $~S_0=\frac l2\,\cos(\theta)~$

you can obtain the velocity $~\dot h(t)~$ and the position $~h(t)=\int \dot h\,dt$

Notice

$~S(t)\le l$

you have one degree on freedom which is $~S~$

you have one degree on freedom which is $~S~$

from the constraint equation

$$ h^2+\left(\frac S2\right)^2=\left(\frac l2\right)^2\tag 1$$

you obtain

$$h=h(S)=\frac 12\sqrt{l^2-S^2}$$

thus $$\dot h=\frac{\partial h}{\partial S}\,\dot S= -\frac 12\,{\frac {S}{\sqrt {-{S}^{2}+{l}^{2}}}}\,\dot S$$

with $$\dot S=f(t)\quad ,S=\int f(t)\,dt+S_0$$

and $~S_0=\frac l2\,\cos(\theta)~$

you can obtain the velocity $~\dot h(t)~$ and the position $~h(t)=\int \dot h\,dt$