Question

two cars are driving at the same constant speed v around a racetrack. however, they are traveling through turns that have different radii, as shown in the drawing. which statement is true about the magnitude of the centripetal acceleration of each car?

the magnitude of the centripetal acceleration of each car is the same, since the cars are moving at the same speed.
the magnitude of the centripetal acceleration of the car at a is less than that of the car at b, since the radius of the circular track is smaller at a.
the magnitude of the centripetal acceleration of the car at a is greater than that of the car at b, since the radius of the circular track is smaller at a.
the magnitude of the centripetal acceleration of the car at a is greater than that of the car at b, since the radius of the circular track is greater at a.

Explanation:

Step1: Recall centripetal - acceleration formula

The formula for centripetal acceleration is $a_c=\frac{v^{2}}{r}$, where $v$ is the speed and $r$ is the radius of the circular path.

Step2: Analyze the effect of radius on centripetal acceleration

Given that the speed $v$ of both cars is the same. From the formula $a_c=\frac{v^{2}}{r}$, we can see that centripetal acceleration $a_c$ is inversely - proportional to the radius $r$ (when $v$ is constant). That is, $a_c\propto\frac{1}{r}$.

Step3: Compare the radii of the two turns

From the drawing, the radius of the turn at A is smaller than the radius of the turn at B, i.e., $r_A

Step4: Determine the centripetal - acceleration relationship

Since $a_c=\frac{v^{2}}{r}$ and $v$ is constant and $r_A < r_B$, then $a_{cA}>a_{cB}$.

Answer:

The magnitude of the centripetal acceleration of the car at A is greater than that of the car at B, since the radius of the circular track is smaller at A.