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?
To find the coefficient of kinetic friction between the tires and the wet road, we can use the equations of motion and the concept of friction.
First, let's list down the given information:
- Mass of the car (m) = 2570.0 kg
- Initial velocity (u) = 14.0 m/s
- Final velocity (v) = 0 m/s (since the car comes to a halt)
- Distance covered (s) = 24.6 m
The key concept here is that when the car is skidding to a halt, the force of kinetic friction acts in the opposite direction to its motion. The magnitude of this force can be calculated using Newton's second law:
F = m * a
In this case, the acceleration (a) is negative since it opposes the car's motion. Rearranging the equation, we have:
F = -m * a
The force of kinetic friction (F) can also be written as:
F = μ * N
where μ is the coefficient of kinetic friction and N is the normal force acting on the car.
Now, let's calculate the normal force (N). In this scenario, the car is on a level road, so the normal force is equal to the weight of the car (mg), where g is the acceleration due to gravity (approximately 9.8 m/s²).
N = mg
Next, we can equate the equations for force of kinetic friction (F) and normal force (N):
F = μN
-ma = μmg
Since the car comes to a halt, its final velocity (v) is 0 m/s. Using the equation of uniformly accelerated motion:
v² = u² + 2as
we can solve for the acceleration (a):
0² = 14.0² + 2a(24.6)
0 = 196.0 + 49.2a
49.2a = -196.0
a = -4.0 m/s²
Substituting this value of acceleration into the equation -ma = μmg, we have:
-(-4.0) * 2570.0 = μ * 2570.0 * 9.8
mu = 1.6
Therefore, the coefficient of kinetic friction between the tires and the wet road is approximately 1.6.