A 820-{\rm kg} sports car collides into the rear end of a 2700-{\rm kg} SUV stopped at a red light. The bumpers lock, the brakes are locked, and the two cars skid forward 3.0 {\rm m} before stopping. The police officer, estimating the coefficient of kinetic friction between tires and road to be 0.80, calculates the speed of the sports car at impact.
14.22 m/s
To find the speed of the sports car at impact, we can use the principle of conservation of momentum. The momentum before the collision is equal to the momentum after the collision.
Let's assume that the initial velocity of the SUV is 0 m/s because it is stopped at a red light.
The momentum before the collision is given by:
Initial momentum = Final momentum
The momentum for an object is calculated by multiplying its mass by its velocity.
For the sports car: Initial momentum = (mass of the sports car) × (initial velocity of the sports car)
For the SUV: Initial momentum = (mass of the SUV) × (initial velocity of the SUV)
Since the initial velocity of the SUV is 0 m/s, the momentum of the SUV before the collision is also 0.
So, we can rewrite the equation as:
(mass of the sports car) × (initial velocity of the sports car) = 0 + (mass of the sports car + mass of the SUV) × (final velocity of the combined cars)
Let's substitute the given values into the equation:
(820 kg) × (initial velocity of the sports car) = (820 kg + 2700 kg) × (final velocity of the combined cars)
Now, let's solve for the final velocity of the combined cars:
(final velocity of the combined cars) = (820 kg) × (initial velocity of the sports car) / (820 kg + 2700 kg)
Given that the two cars skid forward 3.0 m before stopping, we can use the definition of kinetic friction to find the initial velocity of the sports car.
The work done by friction is given by the equation:
Work done by friction = Force of friction × Distance
The force of friction can be calculated using the formula:
Force of friction = Coefficient of kinetic friction × Normal force
The normal force is equal to the weight of the car, which is given by:
Weight = mass × acceleration due to gravity
Substituting the given values into the equation:
Force of friction = (coefficient of kinetic friction) × (mass of the sports car + mass of the SUV) × (acceleration due to gravity)
Lastly, we can use the equation for work done by friction to calculate the initial velocity of the sports car:
Work done by friction = (Force of friction) × (Distance)
The work done by friction is equal to the change in kinetic energy of the sports car:
Work done by friction = (1/2) × (mass of the sports car) × (initial velocity of the sports car)^2
Setting these two equations equal to each other, we can solve for the initial velocity of the sports car.