A 2110 kg car traveling at 10.7 m/s collides with a 2780 kg car that is initially at rest at the stoplight. The cars stick together and move 2.00 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.

Question

A 2110 kg car traveling at 10.7 m/s collides with a 2780 kg car that is initially at rest at the stoplight. The cars stick together and move 2.00 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.

Answer 1

To determine the coefficient of kinetic friction between the cars and the road, we can use the principles of conservation of momentum and the equations of motion.

1. First, let's find the initial momentum of the system before the collision:
Momentum (p) = mass (m) × velocity (v)
For the 2110 kg car: p1 = 2110 kg × 10.7 m/s
For the 2780 kg car at rest: p2 = 0 (as it has no initial velocity)

2. By the principle of conservation of momentum, the total momentum before the collision (p1 + p2) is equal to the total momentum after the collision:
Total initial momentum = Total final momentum
p1 + p2 = (m1 + m2) × V (where V is the final velocity of both cars stuck together)

3. We also know that the negative acceleration during the stopping period is due to friction. The frictional force (f) can be calculated using Newton's second law:
f = mass × acceleration

4. The frictional force (f) is also given by the product of the coefficient of kinetic friction (μ) and the normal force (N):
f = μ × N
The normal force (N) is equal to the weight of the system, which is the sum of the weights of both cars:
N = (m1 + m2) × g (where g is the acceleration due to gravity)

5. The negative acceleration can be found using the equation of motion:
Final velocity squared = Initial velocity squared + 2 × acceleration × distance
V^2 = 0 + 2 × (-a) × 2.00 m (since the cars move 2.00 m before stopping)

Now we have enough information to calculate the coefficient of kinetic friction:

Step 1: Calculate the initial momentum of the system:
p1 = 2110 kg × 10.7 m/s = ...

Step 2: Apply the conservation of momentum:
p1 + p2 = (m1 + m2) × V

Step 3: Calculate the frictional force:
f = μ × N
N = (m1 + m2) × g

Step 4: Calculate the negative acceleration:
V^2 = 0 + 2 × (-a) × 2.00 m

Step 5: Solve for the coefficient of kinetic friction (μ):
μ = f / N

Plug in the given values and solve the equations to find the coefficient of kinetic friction.