Final answer:
To determine how far the car will coast before starting to roll back down the slope, we need to consider the forces acting on the car. The force parallel to the slope is given by F = mg sin(θ), where F is the force, m is the mass of the car, g is the acceleration due to gravity, and θ is the angle of the slope. The force perpendicular to the slope is given by N = mg cos(θ), where N is the normal force, which is equal to the weight of the car.
Step-by-step explanation:
To determine how far the car will coast before starting to roll back down the slope, we need to consider the forces acting on the car. When the car runs out of gas, the only force acting on it is gravity. The force of gravity can be split into two components: one parallel to the slope, which pulls the car down the slope, and one perpendicular to the slope, which pushes the car into the slope.
The force parallel to the slope is given by the equation F = mg sin(θ), where F is the force, m is the mass of the car, g is the acceleration due to gravity, and θ is the angle of the slope. The force perpendicular to the slope is given by the equation N = mg cos(θ), where N is the normal force, which is equal to the weight of the car.
Since the car is not allowed to slip during the coast, the force of static friction acting on the car is equal to the force parallel to the slope. Therefore, we can set F = μsN, where μs is the coefficient of static friction. Rearranging this equation, we get μsN = mg sin(θ). Since N = mg cos(θ), we can substitute this into the equation to get μsmg cos(θ) = mg sin(θ).
Canceling out the mass (m) on both sides of the equation, we are left with the equation μs cos(θ) = sin(θ). We can rearrange this equation to solve for θ: tan(θ) = μs. Substituting the given values into this equation, we get tan(6°) = μs.
Solving this equation using a calculator, we find that μs ≈ 0.1051. Therefore, the car can coast for a distance equal to the distance covered by the car traveling at 26 m/s before it started to roll back down the slope.