A 1970 kg car traveling at 13.8 m/s collides with a 2960 kg car that is initally at rest at a stoplight. The cars stick together and move 1.91 m before friction causes them to stop. Determine the coefficient of kinetic friction between the cars and the road, assuming that the negative acceleration is constant and 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 need to analyze the forces acting on the cars before they come to a stop.
First, let's calculate the initial momentum of the 1970 kg car:
Initial momentum = mass × velocity
Initial momentum of the 1970 kg car = 1970 kg × 13.8 m/s
Now, the 2960 kg car is initially at rest, so its initial momentum is zero (since velocity is zero).
Since the cars stick together and move 1.91 m before stopping, we can calculate the final velocity of both cars using the principle of conservation of momentum:
Total momentum before collision = Total momentum after collision
Since the cars stick together and move as one object after the collision, the final momentum is the sum of their masses multiplied by their final velocity. Let's call the final velocity of both cars after collision "Vf."
Total initial momentum = Total final momentum
(1970 kg × 13.8 m/s) + (2960 kg × 0 m/s) = (1970 kg + 2960 kg) × Vf
Now we can solve for Vf.
Next, we need to determine the deceleration (negative acceleration) that caused the cars to come to a stop. We can use the kinematic equation:
Vf^2 = Vi^2 + 2as
where Vf is the final velocity (zero in this case), Vi is the initial velocity (given as 0 m/s for the second car), a is the acceleration, and s is the stopping distance (1.91 m). We can rearrange the equation to solve for the acceleration:
a = (Vf^2 - Vi^2) / (2s)
Substituting the values we have:
a = (0 - 13.8^2) / (2 × 1.91)
Now that we have the deceleration, we can calculate the frictional force acting on the cars.
Frictional force = mass × acceleration
Let's use the mass of the 2960 kg car to calculate the force.
Finally, we can calculate the coefficient of kinetic friction using the frictional force and the weight of the cars.
Coefficient of kinetic friction = frictional force / weight
The weight of the car is given by:
Weight = mass × acceleration due to gravity
Substituting the values we have, we can now calculate the coefficient of kinetic friction between the cars and the road.