A 0.90 × 10^3 kg sports car collides into the rear end of a 2.4 × 10^3 kg SUV stopped at a red light. The bumpers lock, the brakes are locked, and the two cars skid forward 4.6 m before stopping. The police officer, estimating the coefficient of kinetic friction between tires and road to be 0.40, calculates the speed of the sports car at impact. What was that speed?
The sliding distance can be used to compute the kinetic energy of the joined cars.
(friction force)*X = Kinetic energy at the beginning of skid.
From the KE and masses, get the initial momentum of the joined cars.
Once you have the momentum of the joined cars, you know that it equals the momentum of the sports car before collision.
To find the speed of the sports car at impact, we can use the principles of conservation of momentum and the work-energy principle.
First, let's calculate the initial momentum of the system. The initial momentum is given by the product of the mass of the sports car and its initial velocity, and the mass of the SUV and its initial velocity. Since the SUV is stopped at a red light, its initial velocity is 0.
Initial momentum of the system = (mass of sports car) × (initial velocity of sports car) + (mass of SUV) × (initial velocity of SUV)
= (0.90 × 10^3 kg) × (initial velocity of sports car) + (2.4 × 10^3 kg) × 0
Since the initial velocity of the SUV is 0, the initial momentum of the system simplifies to:
Initial momentum of the system = (0.90 × 10^3 kg) × (initial velocity of sports car)
Now, let's analyze the work-energy principle. The work done by the friction force between the tires and the road causes a loss in the system's kinetic energy. The work-energy principle states that the net work done on an object is equal to the change in its kinetic energy.
The net work done on the system is equal to the work done by the friction force, given by:
Friction force × distance = Work
The friction force can be calculated using:
Friction force = coefficient of kinetic friction × normal force
Using the weight equation (weight = mass × gravity) and assuming a constant gravitational acceleration of 9.8 m/s^2, the normal force acting on the system can be calculated as:
Normal force = (mass of sports car + mass of SUV) × gravity
With the normal force and the coefficient of kinetic friction, we can now calculate the friction force. Then, using the friction force and the given distance, we can calculate the work done on the system.
Since the work done by friction causes the system to lose kinetic energy, the change in the system's kinetic energy is equal to the negative of the work done on the system:
Change in kinetic energy = -Work
We can now equate the change in kinetic energy to the initial momentum of the system to solve for the initial velocity of the sports car.
Change in kinetic energy = -Work
(1/2) × (mass of sports car + mass of SUV) × (initial velocity of sports car)^2 = -Work
Now, let's plug in the known values into the equation and solve for the initial velocity of the sports car.