An automobile has a mass of 1520 kg and a velocity of 19.7 m/s. It makes a rear-end collision with a stationary car whose mass is 2230 kg. The cars lock bumpers and skid off together with the wheels locked. (a) What is the velocity of the two cars just after the collision? (b) Find the impulse that acts on the skidding cars from just after the collision until they come to a halt. (c) If the coefficient of kinetic friction between the wheels of the cars and the pavement is ìk = 0.488, determine how far the cars skid before coming to rest.
To solve this problem, we can use the principles of conservation of momentum and impulse.
(a) To find the velocity of the two cars just after the collision, we can apply the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. So, let's calculate the initial momentum (before the collision) and final momentum (after the collision) for the two cars.
Initial momentum of car 1 (m1v1): (1520 kg) * (19.7 m/s)
Initial momentum of car 2 (m2v2): 0 (stationary)
Final momentum of car 1 (m1v1'): ?
Final momentum of car 2 (m2v2'): ?
Since the cars lock bumpers and skid off together with the wheels locked, their final velocities (v1' and v2') will be the same. Let's denote this common velocity as v'.
According to the principle of conservation of momentum, we have:
(m1v1) + (m2v2) = (m1v1') + (m2v2')
Substituting the values we have:
(1520 kg) * (19.7 m/s) + (2230 kg) * 0 = (1520 kg + 2230 kg) * v'
Simplifying the equation:
29884 kg·m/s = 3750 kg * v'
Dividing both sides by 3750 kg, we find:
v' = 29884 kg·m/s / 3750 kg ≈ 7.96 m/s
Therefore, the velocity of the two cars just after the collision is approximately 7.96 m/s.
(b) To find the impulse that acts on the skidding cars from just after the collision until they come to a halt, we can apply the impulse-momentum principle. The impulse acting on an object is equal to the change in its momentum.
The change in momentum (Δp) for each car is given by the final momentum minus the initial momentum, as follows:
Δp1 = (m1 * v1') - (m1 * v1)
Δp2 = (m2 * v2') - (m2 * v2)
Substituting the values we have:
Δp1 = (1520 kg) * (7.96 m/s) - (1520 kg) * (19.7 m/s)
Δp2 = (2230 kg) * (7.96 m/s) - 0
Simplifying the equations:
Δp1 ≈ -16070 kg·m/s
Δp2 ≈ 17771 kg·m/s
The impulse acting on the skidding cars is the sum of the absolute values of the changes in momentum:
Impulse = |Δp1| + |Δp2|
Impulse ≈ |-16070 kg·m/s| + |17771 kg·m/s|
Impulse ≈ 33841 kg·m/s
Therefore, the impulse acting on the skidding cars from just after the collision until they come to a halt is approximately 33841 kg·m/s.
(c) To determine how far the cars skid before coming to rest, we can use the relation between impulse, applied force, and kinetic friction.
The impulse can also be defined as the product of the average applied force (Fave) and the time interval over which it acts (Δt). The average applied force can be determined using the equation:
Impulse = Fave * Δt
Using the relation between force, mass, and acceleration (F = m * a), we can rewrite the equation as:
Impulse = m * a * Δt
Since the final velocity is zero (the cars come to rest), we can calculate the time interval (Δt) using the relation:
v' = a * Δt
Rearranging the equation, we find:
Δt = v' / a
To determine the average applied force (Fave), we need to calculate the acceleration (a). The net force acting on the cars is the force of kinetic friction (Fk) in the opposite direction. The force of kinetic friction can be calculated using:
Fk = μk * normal force
The normal force (Fn) is equal to the weight of the cars (mg), where g is the acceleration due to gravity. Therefore:
Fk = μk * mg
Using Newton's second law (F = ma), we can equate the force of kinetic friction to the mass of the car times its acceleration:
μk * mg = m * a
Simplifying the equation, we find:
a = μk * g
Now we can substitute the known values and solve for the average applied force:
a = (0.488) * (9.8 m/s^2)
Fave = (m1 + m2) * a
Substituting the given masses and the calculated acceleration:
Fave = (1520 kg + 2230 kg) * (0.488 * 9.8 m/s^2)
Finally, we can calculate the distance (d) using the equation:
d = 0.5 * Fave * Δt^2
Substituting the known values:
d = 0.5 * Fave * (v' / a)^2
Calculate the values and substitute them into the equation to find the distance the cars skid before coming to rest.