A typical running track is an oval with 74-m-diameter half circles at each end. A runner going once around the track covers a distance of 400m . Suppose a runner, moving at a constant speed, goes once around the track in 1 min 40 s. What is her centripetal acceleration during the turn at each end of the track?

## Answer

## General guidance

**Centripetal acceleration:**
Distance refers to “how much ground an object has covered during its motion”. Distance is a scalar quantity. Displacement is a vector quantity that refers to “how far out of place an object is”.
Speed of an object is the “rate of change of distance moved by the object”. It is a scalar quantity. Velocity is a vector quantity. It has both magnitude and direction. Velocity of an object is the rate of change of displacement. Acceleration of an object is defined as the “rate of change of velocity”. It is also a vector quantity.
Speed of the person is given as,
Here, *v* is the speed, *x* is the distance covered, and *t* is the time.
Force may be defined as the sudden push or pull which is capable of changing the direction of the object.
Newton’s second law states that the acceleration of an object produced by a net force is directly proportional to the magnitude of net force and inversely proportional to the mass of the object.
Express the Newton’s second law of motion.
Here, *F* is the force acting on the object, *m* is the mass, and *a* is the acceleration.
Centripetal force may be defined as the “force acting on a body moving in a circular path and directed towards the center around which the body is moving”.
Mathematical expression for centripetal force is,
Here, is the centripetal force, *m* is the mass of the particle, *v* is the velocity and *r* is the radius of the circular orbit.
The acceleration of the particle at the instant during the circular motion is known as the centripetal acceleration. The direction of the acceleration is towards the center of the circle. It depends upon the speed of the particle and the radius of the circular path movement.

## Step-by-step

### Step 1 of 2

Velocity of the runner is given by,
Substitute 400 m for *x* and for .

The time given in the question is . The conversion of time can be carried as,

### Step 2 of 2

From the Newton’s second law of motion,
The centripetal force is given as,
For an object that moves in a circular motion, the force acting is centripetal force. Hence comparing the equations and substitute *ma* for *F*_{C} in equation ,
Radius of the path,
Here, *D* is the diameter of the track.
Substitute for.
Centripetal acceleration is,
Substitute for *v* and 37 m for *r*.

The centripetal acceleration during the turn at each end of the track is .

In centripetal acceleration, the speed remains constant and only the direction of motion alone changes.

### Answer

The centripetal acceleration during the turn at each end of the track is .

### Answer only

The centripetal acceleration during the turn at each end of the track is .