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Circular motion is a fascinating concept that involves the motion of an object along a circular path. In the context of the sprinter, knowing that the velocity is of constant magnitude means the sprinter is maintaining the same speed throughout the turn, but the direction of motion is continuously changing, forming a circle. A key aspect to understand about circular motion is that even if the speed is constant, the velocity is not, because velocity is a vector quantity—which means it has both magnitude and direction. This changing direction requires a net force, directed towards the center of the circle, called the centripetal force. Centripetal acceleration, which is the key aspect of circular motion, indicates how much the direction of the velocity is changing to keep the sprinter moving in a circle, without changing the speed. It is given by the formula: \[ a_c = \frac{v^2}{r} \] Here, \( v \) is the constant speed and \( r \) is the radius of the circular path. Understanding this concept lets us solve for various parameters of circular motion, like the track radius and the period of motion.

The track radius is a critical factor in determining the dimensions of the circular path that the sprinter is running along. It essentially defines how wide the circle is. A larger radius means a bigger circle and a smaller radius leads to a tighter circle.From our problem, we know that the track radius can be derived from the centripetal acceleration formula: \[ r = \frac{v^2}{a_c} \]Here, \( v \) represents the sprinter's velocity, and \( a_c \) is the centripetal acceleration. By substituting the known values, you can find the radius of the track:- Given: - Velocity, \( v = 9.20 \text{ m/s} \) - Centripetal acceleration, \( a_c = 3.80 \text{ m/s}^2 \)- Calculation: - \( r = \frac{(9.20)^2}{3.80} \approx 22.27 \text{ meters} \)This calculation shows that the radius of the track is approximately 22.27 meters. Understanding how to derive the radius from the basic motion parameters underscores how interconnected different aspects of circular motion are.

The period of motion, in the context of circular movement, is defined as the time it takes for an object to complete one full revolution along its circular path. It is a useful measure in understanding the dynamics of circular motion as it relates to both speed and radius.To calculate the period of motion, we use the formula: \[ T = \frac{2\pi r}{v} \]Where \( r \) is the track radius and \( v \) is the constant velocity. In our exercise, we've already determined both values:- Track radius, \( r = 22.27 \text{ meters} \)- Velocity, \( v = 9.20 \text{ m/s} \)When we plug these into the formula, we find: \[ T = \frac{2\pi \times 22.27}{9.20} \approx 15.22 \text{ seconds} \]This means the sprinter takes approximately 15.22 seconds to complete one full lap of the track. Grasping the concept of period is crucial for predicting timing and synchronization in scenarios involving circular motion.