You are driving a 2590.0 kg car at a constant speed of 14.0 m/s along an icy, but straight, level road. As you approach an intersection, the traffic light turns red. You slam on the brakes. Your wheels lock, the tires begin skidding, and the car slides to a halt in a distance of 25.9 m. What is the coefficient of kinetic friction between your tires and the icy road?

To determine the coefficient of kinetic friction between the tires of your car and the icy road, we can use the following equation:

frictional force = normal force * coefficient of kinetic friction.

In this case, the frictional force is necessary to bring the car to a stop, while the normal force is the force exerted by the ground on the car. The normal force can be calculated by:

normal force = mass * gravity,

where mass is the mass of your car (2590.0 kg) and gravity is the acceleration due to gravity (approximately 9.8 m/s^2).

The frictional force can be calculated by multiplying the mass of the car by its deceleration, according to Newton's second law of motion:

frictional force = mass * deceleration,

where deceleration = change in velocity / time taken.

In this case, the change in velocity (Δv) is the initial velocity (14.0 m/s) since the car comes to a complete stop, and the time taken (t) is not given. However, we can solve for time using the formula for distance:

distance = (initial velocity * t) + (0.5 * deceleration * t^2),

where the distance is the stopping distance (25.9 m), the initial velocity is the velocity before braking (14.0 m/s), and deceleration is the rate at which the car slows down.

Rearranging the equation and solving for t:

(0.5 * deceleration * t^2) + (initial velocity * t) - distance = 0.

This is a quadratic equation that can be solved using the quadratic formula. Solving for t will give us the time taken to come to a stop.

Once we have the time taken and the deceleration, we can calculate the frictional force. Rearranging the equation for deceleration:

deceleration = change in velocity / time taken,

we can substitute the values to calculate deceleration.

Finally, substituting the values for mass and deceleration into the equation for frictional force, and using the determined normal force, we can calculate the coefficient of kinetic friction by rearranging the equation:

coefficient of kinetic friction = frictional force / normal force.

By following these steps, you should be able to determine the coefficient of kinetic friction between your tires and the icy road.