Problem 62 - Fundamentals of Physics

Question

What is the magnitude of the acceleration of a sprinter running at \(10 \mathrm{~m} / \mathrm{s}\) when rounding a turn of radius \(25 \mathrm{~m}\) ?

Step-by-Step Solution

Answer

The magnitude of the acceleration is \(4 \mathrm{~m/s^2}\).

Step 1: Understand the Problem

The exercise asks for the magnitude of the acceleration of a sprinter who is rounding a turn. The velocity of the sprinter is given as \(10 \mathrm{~m/s}\), and the radius of the turn is \(25 \mathrm{~m}\). This requires calculating the centripetal acceleration that keeps the sprinter moving in a circular path.

Step 2: Recall the Formula for Centripetal Acceleration

The formula to calculate the centripetal acceleration \(a_c\) is given by: \[ a_c = \frac{v^2}{r} \]where \(v\) is the velocity and \(r\) is the radius of the circular path.

Step 3: Substitute Known Values

Substitute the known values into the formula: \[ v = 10 \mathrm{~m/s} \] and \[ r = 25 \mathrm{~m} \].Thus, the formula becomes: \[ a_c = \frac{(10 \mathrm{~m/s})^2}{25 \mathrm{~m}} \].

Step 4: Perform the Calculation

Calculate the value:\[ a_c = \frac{100 \mathrm{~m^2/s^2}}{25 \mathrm{~m}} = 4 \mathrm{~m/s^2} \].

Step 5: Interpret the Result

The computed result means that the magnitude of the acceleration required to keep the sprinter on the curved path is \(4 \mathrm{~m/s^2}\).

Key Concepts

Uniform Circular Motion

Uniform circular motion refers to the motion of an object traveling at a constant speed along a circular path. In this scenario, although the speed of the object remains unchanged, its direction continually changes as it moves along the circle. This continuous change in direction brings about a concept called centripetal acceleration.

The centripetal acceleration is always directed towards the center of the circle. This is essential for the object to maintain its circular path. In the case of a sprinter rounding a track, this inward acceleration keeps them on the curve, preventing them from moving off in a straight line. Therefore, while the sprinter's speed remains constant, the need to constantly change direction to stay on the track results in the experience of centripetal acceleration.

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without taking into account the forces that cause the motion. It is fundamental for understanding changes in the position, velocity, and acceleration of an object over time.

When discussing circular motion, kinematics helps in understanding how velocity and acceleration interact. In uniform circular motion, the speed is constant, but the velocity, which is a vector, changes due to the continuous change in direction. This is why, even with constant speed, one experiences acceleration — specifically, centripetal acceleration.

• Velocity: Defined as speed with a direction. In circular motion, although the magnitude remains the same, the direction shifts continuously.
• Acceleration: In this case, centripetal acceleration is of interest. Though the magnitude measures the rate of change of speed in linear motion, it measures the rate of change of the velocity's direction in circular motion.

Kinematics provides the foundation for calculating how fast an object needs to accelerate towards the center of its circular path to maintain its motion.

Circular Motion Calculations

Circular motion calculations are crucial to determine the forces and accelerations involved in circular paths. These calculations involve formulas like the one used to determine centripetal acceleration: \[ a_c = \frac{v^2}{r} \]Here, \( a_c \) is the centripetal acceleration, \( v \) is the velocity, and \( r \) is the radius of the circular path.

This formula indicates that:

• Acceleration is directly proportional to the square of the velocity. Thus, the faster an object moves, the higher the required centripetal acceleration to maintain its circular path.
• Acceleration is inversely proportional to the radius. Hence, a tighter curve (smaller radius) demands higher centripetal acceleration compared to a wider turn.

In our problem, understanding these relations helped us calculate that a sprinter moving at 10 m/s around a track with a 25 m radius requires an acceleration of 4 \( \mathrm{m/s^2} \) to stay on the path. These principles are not only applicable in sports but are crucial in engineering and designing roads, roller coasters, and even planetary orbits.