91Ó°ÊÓ

Acceleration is a measure of how quickly an object's velocity changes with time. In circular motion, the acceleration of an object moving at a constant speed in a circle is directed towards the center of the circle. This type of acceleration is called centripetal acceleration.

Centripetal acceleration can be calculated using the formula: \( a = \frac{v^2}{r} \), where \(v\) is the speed of the object, and \(r\) is the radius of the circle.
To solve the exercise given, we used the provided values

• Speed of the sprinter, \(v = 10 \, m/s\)
• Radius of the turn, \(r = 25 \, m\)

We substituted these values into the centripetal acceleration formula:
\( a = \frac{(10 \, m/s)^2}{25 \, m} \)
Performing the calculation, we get:
\( a = \frac{100 \, m^2/s^2}{25 \, m} = 4 \, m/s^2 \)

Thus, the magnitude of the acceleration is \( 4 \, m/s^2 \).

Kinematic equations describe the motion of objects without considering the forces causing the motion. In the case of circular motion, these equations are slightly modified to account for the constant change in direction.

Some of the key kinematic equations in linear motion are:

• \( v = u + at \)
• \( s = ut + \frac{1}{2}at^2 \)
• \( v^2 = u^2 + 2as \)

Here, \(u\) represents initial velocity, \(v\) is final velocity, \(a\) is acceleration, \(t\) is time, and \(s\) is displacement.

For circular motion, the equations incorporate angular parameters. One main equation related to the exercise is the one for centripetal acceleration, which is derived from transforming linear acceleration concepts to circular pathways. This is crucial in understanding how objects move in a curved path.

Circular motion refers to the movement of an object along the perimeter of a circle or a circular path. It can be uniform, where the velocity remains constant, or non-uniform, where the velocity changes. The type of motion described in the exercise is uniform circular motion since the sprinter is moving at a constant speed.

Key concepts in circular motion include:

• Centripetal Force: The force directed towards the center of the circle keeping the object in motion along the circular path.
• Centripetal Acceleration: As discussed, it is the acceleration directed towards the center of the circle, calculated using \( a = \frac{v^2}{r} \).
• Period and Frequency: The time taken to complete one full circle (period) and the number of circles completed per unit time (frequency).

Understanding these concepts helps us to grasp how the sprinter maintains their trajectory while rounding the turn, providing a clear connection between speed, radius, and the required centripetal acceleration.