7. (15 points) A rigid rod is moving on a horizontal table. At a certain moment, the speed of one end of the rod is \( v_{1}=5 \mathrm{~m} / \mathrm{c} \), and the speed of the other end is \( v_{2}=4 \) m / s, directed along the axis of the rod (see figure). Determine the speed of the midpoint of the rod at this moment.

![](https://cdn.mathpix.com/cropped/2024_05_06_b6ca313a2688e0b12313g-04.jpg?height=157&width=842&top_left_y=2106&top_left_x=684)
Answer: $\approx 4.3 \mathrm{~m} / \mathrm{s}$.

Solution. All points on the rod have a velocity component directed to the right, equal to $v_{2}$.

(4 points)

Therefore, the velocity component directed upwards for point A: $v_{\text {vert A }}=\sqrt{v_{1}^{2}-v_{2}^{2}}=3 \mathrm{m} / \mathrm{s}$.

The velocity component of point $B$, directed upwards:

$v_{\text {vert B}}=\frac{1}{2} v_{\text {vert}}=1.5 \mathrm{~m} / \mathrm{s}$.

The total velocity of point $B: v=\sqrt{v_{2}^{2}+v_{\text {vert}}^{2}}=\sqrt{4^{2}+1.5^{2}} \approx 4.3 \mathrm{~m} /$ s. (3 points)