You are driving a 2570.0-kg car at a constant speed of 14.0 m/s along a wet, but straight, level road. As you approach an intersection, the traffic light turns red. You slam on the brakes. The car's wheels lock, the tires begin skidding, and the car slides to a halt in a distance of 24.6 m. What is the coefficient of kinetic friction between your tires and the wet road?
Friction work = initial kinetic energy
M*g*mu*X = (1/2)M*V^2
M cancels out. Solve for mu, the coefficient of friction
To find the coefficient of kinetic friction between your tires and the wet road, we can use the equation:
μ = (m*g) / (m*(a + g))
Where:
μ = coefficient of kinetic friction
m = mass of the car (2570.0 kg)
g = acceleration due to gravity (9.8 m/s²)
a = acceleration
First, we need to find the acceleration of the car by using the kinematic equation:
(vf² - vi²) = 2aΔx
Where:
vi = initial velocity (14.0 m/s)
vf = final velocity (0 m/s)
Δx = displacement (24.6 m)
Rearranging the equation and inserting the given values:
a = (vf² - vi²) / (2Δx)
= (0² - 14.0²) / (2 * 24.6)
≈ -3.53 m/s² (negative because the car is decelerating)
Now we can substitute the values into the coefficient of kinetic friction equation:
μ = (m*g) / (m*(a + g))
= (2570.0 * 9.8) / (2570.0 * (-3.53) + 2570.0 * 9.8)
≈ 0.60
Therefore, the coefficient of kinetic friction between your tires and the wet road is approximately 0.60.
To find the coefficient of kinetic friction between your car's tires and the wet road, we can use the equation that relates the force of kinetic friction to the normal force and the coefficient of kinetic friction:
Frictional force = coefficient of kinetic friction * normal force
First, we need to find the normal force acting on the car. The normal force is the force exerted by the road on the car in the upward direction to counteract the weight of the car. In this case, the weight of the car is equal to its mass times the acceleration due to gravity (9.8 m/s^2):
Weight of the car = mass of the car * acceleration due to gravity
= 2570.0 kg * 9.8 m/s^2
Next, we can use Newton's second law of motion to find the net force acting on the car as it decelerates to a stop:
Net force = mass of the car * acceleration
Since the car comes to a stop, the net force acting on it is equal to the frictional force. So we can write:
Frictional force = mass of the car * acceleration
Now, we need to find the acceleration of the car. Using the equations of motion, we can relate the distance traveled by the car, its initial velocity, acceleration, and time:
Distance = (Initial velocity * Time) + (0.5 * Acceleration * Time^2)
In this case, the distance traveled by the car is 24.6 m. Since the car comes to a stop (final velocity is 0), the equation simplifies to:
Distance = 0.5 * Acceleration * Time^2
To find the time taken for the car to stop, we can rearrange the equation:
Time = sqrt((2 * Distance) / Acceleration)
Now we have two equations involving the acceleration and the frictional force. We can equate them and solve for the coefficient of kinetic friction:
Frictional force = coefficient of kinetic friction * (mass of the car * acceleration)
mass of the car * acceleration = coefficient of kinetic friction * (mass of the car * acceleration)
The mass of the car cancels out, and we are left with:
Acceleration = coefficient of kinetic friction * acceleration
Simplifying further, we find:
coefficient of kinetic friction = Acceleration / acceleration
To find the acceleration, we can use:
Acceleration = (Final velocity^2 - Initial velocity^2) / (2 * Distance)
Substituting the known values, we can calculate the acceleration.
Finally, we can substitute the calculated acceleration and the given values into the formula for the coefficient of kinetic friction to find the answer.