Final answer:
To find the minimum stopping distance of the automobile, the force of static friction and the mass of the vehicle are used to calculate the deceleration. Then, a kinematic equation relates the initial velocity, deceleration, and stopping distance to solve for the minimum distance required to stop the car on a 15° grade.
Step-by-step explanation:
To calculate the minimum distance needed to stop the car, we first need to consider the forces acting on the vehicle as it goes up the incline. The force of static friction between the tires and the road plays a crucial role, as it provides the maximum deceleration force the tires can apply without slipping. Using the given force of static friction (7960 N) and the mass of the automobile (1200 kg), we can calculate the deceleration using Newton's second law: F = ma, where F is the force applied, m is the mass, and a is the acceleration (or in this case, deceleration).
The deceleration a can be calculated by rearranging the formula to a = F/m. We then use the kinematic equation v² = u² + 2as, where v is the final velocity (0 m/s for a complete stop), u is the initial velocity, a is the deceleration, and s is the stopping distance. By substituting in the values for u, a, and v and solving for s, we can find the minimum stopping distance. However, remember to also account for the gravitational component parallel to the incline because it affects the effective force available for deceleration.