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 apply the principles of Newton's laws of motion and conservation of momentum.
(a) To find the velocity of the two cars just after the collision, we can use 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 calculated by multiplying its mass by its velocity:
Momentum = mass × velocity
Before the collision:
Momentum of Car 1 (m1) = mass of Car 1 × velocity of Car 1
= 1520 kg × 19.7 m/s
Momentum of Car 2 (m2) = mass of Car 2 × velocity of Car 2
= 2230 kg × 0 m/s (since it is stationary)
Total momentum before the collision = m1 + m2
After the collision, the two cars lock bumpers and move together:
Total momentum after the collision = (m1 + m2) × velocity of the combined cars
Since there is no external force acting on the system after the collision, the total momentum is conserved:
m1 × v1 + m2 × v2 = (m1 + m2) × vf
where v1 and v2 are the velocities before the collision, and vf is the velocity of the combined cars after the collision.
Plugging in the values:
(1520 kg × 19.7 m/s) + (2230 kg × 0 m/s) = (1520 kg + 2230 kg) × vf
Solving for vf:
(29864 kg · m/s) = (3750 kg) × vf
vf = 29864 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 use the equation:
Impulse = change in momentum
The change in momentum can be calculated by subtracting the momentum before the collision from the momentum after the collision:
Change in momentum = (m1 + m2) × vf - (1520 kg × 19.7 m/s)
Plugging in the values:
Change in momentum = (3750 kg) × 7.96 m/s - (1520 kg × 19.7 m/s)
Solving for the change in momentum:
Change in momentum = 29925 kg · m/s - 29944 kg · m/s
Change in momentum ≈ -19 kg · m/s
Therefore, the impulse that acts on the skidding cars from just after the collision until they come to a halt is approximately -19 kg · m/s.
(c) If the coefficient of kinetic friction between the wheels of the cars and the pavement is μk = 0.488, we can determine how far the cars skid before coming to rest by using the equation:
Force of kinetic friction = coefficient of kinetic friction × normal force
The normal force can be calculated as the weight of the cars:
Normal force = mass × gravitational acceleration
= (1520 kg + 2230 kg) × 9.8 m/s^2
The force of kinetic friction can be calculated as:
Force of kinetic friction = normal force × coefficient of kinetic friction
Since the force of kinetic friction opposes the motion of the cars, it will be in the opposite direction to the motion.
The work done by friction force to bring the cars to rest is equal to the product of the force of kinetic friction and the distance traveled:
Work done by friction = Force of kinetic friction × distance
Since the work done by friction is equal to the change in kinetic energy of the cars:
Work done by friction = change in kinetic energy = 0.5 × (mass of the cars) × (final velocity)^2
Setting the work done by friction equal to the change in kinetic energy:
Force of kinetic friction × distance = 0.5 × (mass of the cars) × (final velocity)^2
Plugging in the values:
(μk × normal force) × distance = 0.5 × (1520 kg + 2230 kg) × (7.96 m/s)^2
Solving for distance:
distance = (0.5 × (3750 kg) × (7.96 m/s)^2) / (μk × normal force)
Plugging in the values:
distance = (0.5 × 3750 kg × 63.3616 m^2/s^2) / (0.488 × (1520 kg + 2230 kg) × 9.8 m/s^2)
Simplifying:
distance ≈ 144.5 meters
Therefore, the cars skid for approximately 144.5 meters before coming to rest.
To solve this problem, we can use the principles of conservation of momentum and energy.
(a) To find the velocity of the two cars just after the collision, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision when there are no external forces acting on the system.
The initial momentum of the first car (m1) is given by:
Momentum1 = mass1 * velocity1
Momentum1 = 1520 kg * 19.7 m/s
The initial momentum of the second car (m2) is zero since it is stationary.
Total initial momentum = Momentum1 + Momentum2 = Momentum1 + 0
Now, the final momentum is calculated similarly, but considering the velocity of both cars locked together:
Total final momentum = (m1 + m2) * velocity_final
Using the principle of conservation of momentum, we equate the initial and final momentum:
Momentum1 + 0 = (m1 + m2) * velocity_final
Solving for velocity_final, we have:
velocity_final = (Momentum1 + 0) / (m1 + m2)
Substituting the known values, we can calculate the velocity_final.
(b) To find the impulse acting on the skidding cars, we can use the impulse-momentum principle. The impulse exerted on an object is equal to the change in momentum of that object.
Impulse = change in momentum
Since the cars come to a halt, the momentum before the collision is equal to the momentum after the collision.
Initial momentum = (m1 * velocity1) + (m2 * 0)
Final momentum = (m1 + m2) * 0
Change in momentum = Final momentum - Initial momentum
Now, we can calculate the impulse by substituting the known values.
(c) To determine the distance the cars skid before coming to rest, we can use the work-energy principle. The work done by the friction force is equal to the change in kinetic energy.
Work done by friction force = Change in kinetic energy
The change in kinetic energy can be calculated as the initial kinetic energy (0.5 * (m1 + m2) * velocity_final^2) minus the final kinetic energy (0.5 * (m1 + m2) * 0^2).
Since the work done by friction force is equal to the force of friction multiplied by the distance, we can write:
Work done by friction force = force of friction * distance
We can now equate the expressions for work done by friction force and change in kinetic energy, and solve for distance:
force of friction * distance = Change in kinetic energy
Substituting the known values, we can calculate the distance.