A 2160 kg car traveling at 10.5 m/s collides with a 2740 kg car that is initially at rest at the stoplight. The cars stick together and move 4.00 m before friction causes them to stop. Determine the coefficient of kinetic friction betwen the cars and the road, assuming that the negative acceleration is constant and that all wheels on both cars lock at the time of impact.
To determine the coefficient of kinetic friction between the cars and the road, we can use the principles of conservation of momentum and work done by friction.
1. First, let's calculate the initial momentum of the system before the collision. Momentum is calculated by multiplying mass and velocity. The momentum of the first car is given by:
Momentum_car1_before = mass_car1 * velocity_car1
Momentum_car1_before = 2160 kg * 10.5 m/s
2. The momentum of the second car is initially at rest and therefore zero:
Momentum_car2_before = 0
3. The total initial momentum of the system is the sum of the individual momenta:
Total_momentum_before = Momentum_car1_before + Momentum_car2_before
4. After the collision, the two cars stick together and move a distance of 4.00 m before coming to a stop. The work done by friction is equal to the force of friction multiplied by the distance traveled. Thus, the work done by friction is:
Work = force_friction * distance
5. The work done by friction can also be calculated as the change in kinetic energy of the system. Initially, the system has kinetic energy due to the motion of the first car. After the collision, the system comes to rest, so the final kinetic energy is zero. Therefore, the work done by friction is:
Work = change in kinetic energy
6. The change in kinetic energy is given by:
Change in kinetic energy = Final kinetic energy - Initial kinetic energy
7. Since the final kinetic energy is zero and the initial kinetic energy is equal to the initial momentum of the system (using the equation for kinetic energy: KE = (1/2) * mass * velocity^2), the change in kinetic energy can be simplified to:
Change in kinetic energy = -Initial momentum
8. The work done by friction is equal to the change in kinetic energy. Therefore, we can set the work equation equal to the change in kinetic energy equation:
force_friction * distance = -Initial momentum
9. The force of friction can be calculated by rearranging the equation as:
force_friction = -Initial momentum / distance
10. The force of friction can also be expressed as the product of the coefficient of kinetic friction (μ) and the normal force (N). The normal force is the weight of the system, which is the sum of the weights of the two cars (mg). Rearranging the equation, we get:
force_friction = μ * (mass_car1 + mass_car2) * g
11. Setting the two equations for force of friction equal to each other, we can solve for the coefficient of kinetic friction (μ):
-Initial momentum / distance = μ * (mass_car1 + mass_car2) * g
12. Plugging in the given values for mass_car1, mass_car2, velocity_car1, and distance, as well as the acceleration due to gravity (g = 9.8 m/s^2), we can solve for the coefficient of kinetic friction (μ).