Understanding the Ratio of Distance to Displacement Along a Semicircle

The ratio of distance to displacement is a fundamental concept in geometry and physics, particularly when dealing with circular paths. In this article, we will delve into the specific case of moving along a semicircle and explore the ratios involved.

Distance and Displacement in a Semicircle

To understand the relationship between distance and displacement in a semicircle, we need to define two key terms:

Distance: This is the total length of the path traveled. For a semicircle with radius r, the distance traveled along the semicircle is half the circumference of a full circle. Displacement: This is the straight-line distance between the initial and final points. For a semicircle, if you start at one end and move to the other end, the displacement is the diameter of the semicircle.

Calculating the Distance and Displacement

The distance traveled along the semicircle is given by:

Distance (frac{1}{2} times 2pi r pi r)

The displacement is given by:

Displacement (2r)

Now, we can calculate the ratio of distance to displacement:

Ratio (frac{pi r}{2r} frac{pi}{2})

Therefore, the ratio of distance to the magnitude of displacement when moving along a semicircle is (frac{pi}{2}).

Example Problems

Let's consider an example problem to illustrate this concept. Suppose a body travels a distance of 4 meters along a semicircular path.

Example 1

According to the given data, the required magnitude of displacement of the body is the diameter of the semicircular path:

Magnitude of displacement (frac{4 text{ m}}{pi/2} frac{4 text{ m}}{22/7/2} approx frac{4 text{ m} times 14}{22} frac{56 text{ m}}{22} approx 2.545 text{ m})

The required ratio of distance to displacement is:

Ratio (4 : frac{56}{22} 44 : 28 11 : 7)

Example 2

Now, let's assume the radius of the circle is r meters. The distance traveled is given by:

Distance (frac{22/7 times 2 times r}{4} frac{11r}{7})

The displacement is the diameter of the semicircle:

Displacement (2r)

Thus, the required ratio is:

Ratio (frac{11r/7}{2r} frac{11}{14} approx 1.1111)

By considering the unit circle, we also see that the ratio of distance to displacement is (frac{pi}{2}) because the diameter of the semicircle is the diameter of the full circle.

Conclusion

In summary, when moving along a semicircular path, the ratio of distance to the magnitude of displacement is always (frac{pi}{2}). This concept is crucial for understanding the geometry and motion along curved paths.