To find the average velocity given the motion of a person traveling along a straight line with two different velocities, we start by calculating the average velocity ($ v_{\text{avg}} $) using the total distance traveled divided by the total time taken.

The distances and velocities are given as follows:

Distance $ x $ at velocity $ v_1 = 5 \, \text{m/s} $

Distance $ \frac{3}{2}x $ at velocity $ v_2 $

The formula for average velocity is:

$ v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} $

Given:

$ v_{\text{avg}} = \frac{50}{7} \, \text{m/s} $

The total distance traveled is:

$ x + \frac{3x}{2} = \frac{5x}{2} $

The time taken to cover each segment is given by:

Time for $ x $: $ t_1 = \frac{x}{v_1} = \frac{x}{5} $

Time for $ \frac{3x}{2} $: $ t_2 = \frac{\frac{3x}{2}}{v_2} = \frac{3x}{2v_2} $

Now, using the formula for average velocity:

$ \frac{50}{7} = \frac{\frac{5x}{2}}{\frac{x}{5} + \frac{3x}{2v_2}} $

Simplifying the equation:

$ \frac{50}{7} = \frac{5/2}{\frac{1}{5} + \frac{3}{2v_2}} $

Cross-multiplying gives:

$ \frac{1}{5} + \frac{3}{2v_2} = \frac{7}{20} $

Solving for $ \frac{3}{2v_2} $:

$ \frac{3}{2v_2} = \frac{7}{20} - \frac{1}{5} = \frac{7-4}{20} = \frac{3}{20} $

Finally, solving for $ v_2 $:

$ \frac{3}{2v_2} = \frac{3}{20} $

$ v_2 = 10 \, \text{m/s} $

Thus, the value of $ v_2 $ is $ 10 \, \text{m/s} $.