To solve this problem, we need to determine the stopping distance of a car with the following given values:

Given Data:

• Mass of the car, $m = 945 \, \text{kg}$
• Initial speed of the car, $v = 22.6 \, \text{m/s}$
• Coefficient of kinetic friction between the tires and the road, $\mu_k = 0.972$
• The car comes to a stop, so the final velocity, $v_f = 0 \, \text{m/s}$

Solution Steps:

1. Determine the Frictional Force:

   The frictional force $f$ can be calculated using the formula: $f = \mu_k \cdot N$ where $N$ is the normal force. Since the car is on a flat surface, $N = m \cdot g$, where $g = 9.8 \, \text{m/s}^2$ (acceleration due to gravity).

   Substituting the values: $f = \mu_k \cdot m \cdot g = 0.972 \cdot 945 \cdot 9.8$

   Calculating $f$: $f = 9006.036 \, \text{N}$

2. Calculate the Deceleration (Negative Acceleration):

   Since this frictional force is the only horizontal force acting on the car, it provides the deceleration $a$: $f = m \cdot a$ Solving for $a$: $a = \frac{f}{m} = \frac{9006.036}{945}$

   Calculating $a$: $a = 9.528 \, \text{m/s}^2$ This deceleration is in the opposite direction of motion, so we take it as $-9.528 \, \text{m/s}^2$.

3. Calculate the Stopping Distance Using Kinematic Equation:

   We use the kinematic equation: $v_f^2 = v^2 + 2 \cdot a \cdot d$ Rearranging to solve for $d$: $d = \frac{v_f^2 - v^2}{2 \cdot a}$ Substituting $v_f = 0$, $v = 22.6 \, \text{m/s}$, and $a = -9.528 \, \text{m/s}^2$: $d = \frac{0 - (22.6)^2}{2 \cdot (-9.528)}$

   Calculating $d$: $d = \frac{-510.76}{-19.056} \approx 26.8 \, \text{m}$

Final Answer:

The distance required to stop the car is approximately $26.8 \, \text{m}$.

Would you like further clarification on any step or additional details?

Here are five related questions to consider:

1. How would the stopping distance change if the coefficient of friction were lower?
2. What would happen to the stopping distance if the car's mass were doubled?
3. How does an incline affect the stopping distance if the road is sloped?
4. How would a higher initial speed affect the stopping distance?
5. What role does tire condition play in determining the stopping distance?

Tip: Remember, the stopping distance increases with both speed and reduced friction, making it crucial to consider both factors when driving conditions change.