Let's solve the given problem step by step.

Problem Information:

1. Mass of the car, $m = 615 \, \text{kg}$
2. Time taken to complete the circular path, $t = 14.3 \, \text{s}$
3. Radius of the circular track, $r = 50.0 \, \text{m}$

(a) Finding the Acceleration of the Car

Since the car is moving in a circular path, it undergoes centripetal acceleration, which is given by the formula:

$a = \frac{v^2}{r}$

To find $v$ (the speed of the car), we first calculate the circumference of the circular track:

$\text{Circumference} = 2 \pi r = 2 \pi \times 50 = 100 \pi \, \text{m}$

Since the car takes 14.3 seconds to complete one lap, the speed $v$ is:

$v = \frac{\text{Circumference}}{t} = \frac{100 \pi}{14.3} \approx 21.98 \, \text{m/s}$

Now, using the centripetal acceleration formula:

$a = \frac{v^2}{r} = \frac{(21.98)^2}{50} \approx 9.66 \, \text{m/s}^2$

(b) Finding the Average Force

The centripetal force required to keep the car in circular motion is calculated using Newton's second law:

$F = m \times a$

Substituting the values:

$F = 615 \times 9.66 \approx 5943.9 \, \text{N}$

Summary of Answers:

• (a) The acceleration of the car is approximately $9.66 \, \text{m/s}^2$.
• (b) The average force exerted by the track on the car's tires is approximately $5943.9 \, \text{N}$.

Would you like further details on any step, or have any additional questions?

Here are some related questions to expand your understanding:

1. What would happen to the acceleration if the radius of the track were doubled?
2. How does the mass of the car affect the centripetal force needed?
3. What would be the required force if the car traveled at twice the speed?
4. How would the force change if the car completed the circle in half the time?
5. What is the relationship between centripetal force and the speed of the object in circular motion?

Tip: In circular motion, increasing the speed significantly increases the required centripetal force since it’s proportional to the square of the velocity.