A horse rider covers half the distance with $$5$$ m s$$^{-1}$$ speed. The remaining part of the distance was travelled with speed $$10$$ m s$$^{-1}$$ for half the time and with speed $$15$$ m s$$^{-1}$$ for other half of the time. The mean speed of the rider averaged over the whole time of motion is $$\frac{x}{7}$$ m s$$^{-1}$$. The value of $$x$$ is ______.

A horse rider covers half the distance at 5 m/s, then the remaining half distance with speeds 10 and 15 m/s for equal time intervals.

Set up variables.

Let the total distance = $$2d$$.

Time for first half.

$$t_1 = \frac{d}{5}$$

Time for second half.

Let the time for each speed be $$t_2$$. Distance in second half:

$$d = 10t_2 + 15t_2 = 25t_2 \implies t_2 = \frac{d}{25}$$

Total time for second half = $$2t_2 = \frac{2d}{25}$$

Calculate mean speed.

Total time = $$\frac{d}{5} + \frac{2d}{25} = \frac{5d + 2d}{25} = \frac{7d}{25}$$

Mean speed = $$\frac{2d}{\frac{7d}{25}} = \frac{50}{7}$$ m/s

Therefore, $$x = \boxed{50}$$.