A 930 kg sports car collides into the rear end of a 2500 kg SUV stopped at a red light. The bumpers lock, the brakes are locked, and the two cars skid forward 2.6 m before stopping. The police officer, knowing that the coefficient of kinetic friction between tires and road is 0.80, calculates the speed of the sports car at impact.
what was the speed?
Vf^2=Vi^2+2ad
a= frictionforce/mass where friction force is mu*(massbothcars)g
let me check that. Energy wise
1/2 masssportscar*Vi^2=frictionforce*massboth cars*distance
Vi^2= 2* mu(massbothcars)distance/masssportscar
I am glad I checked it.
To find the speed of the sports car at impact, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The momentum (p) is defined as the product of an object's mass and its velocity:
p = mass × velocity
In this case, we have two objects involved in the collision, the sports car and the SUV. Let's denote the mass of the sports car as m1 and its velocity as v1, and the mass of the SUV as m2 and its velocity as v2.
Before the collision, both cars are stationary, so the momentum of each car is zero. Therefore, the total momentum before the collision is also zero.
After the collision, the cars skid forward and come to a stop. We are given the distance they skid (2.6 m) and the coefficient of kinetic friction (0.80), which allows us to calculate the frictional force acting on the cars.
The equation for the frictional force (F) is given by:
F = μ × weight
where μ is the coefficient of kinetic friction and weight is the force due to gravity acting on an object, given by:
weight = mass × gravitational acceleration
In this case, the weight of the car can be calculated as:
weight = mass × 9.8 m/s^2
Now, the frictional force can be related to the deceleration (a) of the car using Newton's second law:
F = mass × acceleration
Simplifying the equation, we have:
μ × weight = mass × acceleration
Solving for the acceleration, we get:
acceleration = μ × weight / mass
Now, using the equation of motion:
final velocity^2 = initial velocity^2 + 2 × acceleration × distance
Because the cars come to a stop, the final velocity is zero. Rearranging the equation, we have:
initial velocity^2 = - 2 × acceleration × distance
Substituting the values given, we have:
initial velocity^2 = - 2 × (μ × weight / mass) × distance
Simplifying further, we get:
initial velocity = sqrt(-2 × (μ × weight / mass) × distance)
Now we can substitute the values into the equation and calculate the initial velocity of the sports car:
mass of the sports car (m1) = 930 kg
mass of the SUV (m2) = 2500 kg
coefficient of kinetic friction (μ) = 0.80
distance traveled (d) = 2.6 m
gravitational acceleration (g) = 9.8 m/s^2
Plugging in these values, we get:
initial velocity = sqrt(-2 × (0.80 × (m1 + m2) × g / m1) × d)
Calculating further using the given values:
initial velocity = sqrt(-2 × (0.80 × (930 + 2500) × 9.8 / 930) × 2.6)
After substituting the values into the equation and evaluating, the initial velocity of the sports car at impact can be found.