During a \(200 \mathrm{~m}\) race, a sprinter is running with a speed of \(10 \mathrm{~m} / \mathrm{s}\) through the first curve. The length of the curve is \(100 \mathrm{~m}\). (a) Find the radius, \(R\), of the curve (it is a perfect half-circle). (b) Find the magnitude and direction of the acceleration \(a\) of the sprinter.

The radius of curvature is essential for understanding the dynamics of circular motion. It represents the radius of the perfect circle that forms the curve of the path. In our exercise, the sprinter runs along a curve that forms a half-circle of 100 meters in length. By knowing the total distance around the circle is given by its circumference formula, which is \[ C = 2\pi R \]. Since the curve is half of the full circle, we set the length of the curve to half the circumference, i.e., \[ 100 \text{ m} = \pi R \]. By solving this equation for R, we find the radius to be \[ R = \frac{100 \text{ m}}{\pi} \], which approximately equals 31.83 meters. The radius plays a critical role in further calculations such as determining centripetal acceleration.

Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed towards the center of the curvature. It can be calculated using the formula \[ a = \frac{v^2}{R} \], where \ v \ is the velocity and \ R \ is the radius of curvature. For the sprinter running at 10 meters per second with a radius of approximately 31.83 meters, the centripetal acceleration is calculated as \[ a = \frac{(10 \text{ m/s})^2}{31.83 \text{ m}} \], yielding about 3.14 meters per second squared. Understanding centripetal acceleration helps in analyzing the forces acting on the sprinter during the curve, ensuring that they can maintain their speed without slipping.

Circular motion dynamics encompasses the study of forces and accelerations when an object moves along a circular path. Key components include centripetal force, which is required to keep an object moving in a circular path, and the centripetal acceleration. The force is calculated as \[ F_c = m \times a \], where \ m \ is the mass of the object and \ a \ is the centripetal acceleration. For our sprinter, the acceleration is always directed towards the center of the path, ensuring a continuous circular motion. Without the correct centripetal force, the sprinter would move off the track due to inertia. It's crucial to understand that every motion in a curved path involves these dynamic factors.

In biomechanics, analyzing a sprinter's motion involves understanding the interplay between speed, the shape of the track, and the forces exerted. When a sprinter runs through a curve, they need to generate sufficient centripetal force to counteract the outward push due to inertia. The radius of the curve determines the required centripetal acceleration and force. Smaller radii demand more significant forces, making it essential to balance speed and biomechanics to prevent injuries. Biomechanical studies help sprinters optimize their posture, foot placement, and force application to smoothly navigate curves and transitions while maintaining speed and efficiency. This understanding directly correlates with performance improvements and injury reduction strategies.