The problem is about a train moving from point A to point B, where the average speed is represented by $r_m$ and the magnitude of the average velocity is represented by $v_m$. We are tasked with determining the correct relationship between these two quantities.

The additional information you provided mentions that the path between A and B can be visualized as a circumference, with the distance between A and B being the diameter.

Let’s break down the problem:

1. Average speed ($r_m$) is the total distance traveled by the train along its path. Since the train is traveling along the circumference, this is the perimeter of a semicircle. If the radius of the circle is given as 30 cm, the perimeter of the semicircular path is: $\text{Perimeter of semicircle} = \pi \times \text{radius} = \pi \times 30 \, \text{cm} = 30\pi \, \text{cm}.$

2. Average velocity ($v_m$) is the straight-line distance between the initial and final points, which is the diameter of the circle: $\text{Diameter} = 2 \times \text{radius} = 2 \times 30 \, \text{cm} = 60 \, \text{cm}.$

Now, to find the ratio $\frac{r_m}{v_m}$:

$\frac{r_m}{v_m} = \frac{30\pi}{60} = \frac{\pi}{2}.$

So, the correct answer is Option A: $\frac{r_m}{v_m} = \frac{\pi}{2}.$

Would you like further clarification on any part of this explanation?

Here are 5 related questions:

1. How would the result change if the path was a full circle instead of a semicircle?
2. What is the difference between speed and velocity in terms of their definitions?
3. How can the concept of average speed apply to a non-circular path?
4. Why is the average velocity based on displacement, not the total path length?
5. What would be the ratio $\frac{r_m}{v_m}$ if the train moved along a quarter of the circle?

Tip:

Remember, speed is a scalar quantity representing how fast an object moves, while velocity is a vector that also includes direction.