Answer
See the detailed answer below.
Work Step by Step
Since the two masses are connected by a pivoting rigid rod, they must have the same speed even if they do not have the same velocity. It means that they will have different directions since one of them is moving vertically and the other is moving horizontally.
As we see in the figure below, the rod represents a hypotenuse of a right triangle. The base of this triangle is $x_2$ which is the distance traveled by $m_2$ while its height is $y_1$ which is the distance traveled by $m_1$.
We can apply the Pythagorean theorem;
$$L^2=x_2^2+y_1^2$$
To find the velocities relations, we need to differentiate this equation relative to $t$.
$$\dfrac{d}{dt}L^2=\dfrac{d}{dt}x_2^2+\dfrac{d}{dt}y_1^2$$
We know that the length of the rigid rod is constant, so the left side must be zero.
$$0=2x_2\overbrace{ \dfrac{dx_2}{dt} }^{v_{x2}} +2y_2 \overbrace{ \dfrac{dy_1}{dt}}^{v_{y1}} $$
$$-2x_2v_{x2}=2y_1v_{y1}$$
Divide both sides by $-2x_2$;
$$ v_{x2}=-\overbrace{ \dfrac{y_1}{x_2}}^{\tan \theta} v_{y1}$$
Therefore,
$$\boxed{ v_{x2}=- v_{y1}\tan\theta}$$
