Final answer:
To calculate the maximum spring compression when two cars couple together, first use the conservation of momentum to find their combined velocity and then apply Hooke's law and the conservation of energy to equate the kinetic energy of the cars to the potential energy in the spring, allowing for the calculation of the spring's compression.
Step-by-step explanation:
To find the maximum compression of the spring when an 18000 kg freight car, which is initially at rest and hit by a 9400 kg car moving at 8.0 m/s, couples together with it, we use the principles of conservation of momentum and the relation of force to spring compression described by Hooke's law.
Step 1: Conservation of Momentum
The initial total momentum is equal to the momentum of the moving car since the freight car is at rest. Using the conservation of momentum before and after the collision, the combined velocity (vcombined) of both cars coupled together can be found by:
Momentumbefore = Momentumafter
(M1 × v1) + (M2 × v2) = (M1 + M2) × vcombined
(9400 kg × 8.0 m/s) + (18000 kg × 0 m/s) = (9400 kg + 18000 kg) × vcombined
Step 2: Hooke's Law
Once the combined velocity is known, it can be used to determine the kinetic energy just before the cars compress the spring. The kinetic energy is then equal to the potential energy stored in the spring at maximum compression, according to the conservation of energy. Hooke's law states that the force on a spring is proportional to its displacement (x):
F = k × x
Where k is the spring constant and x is the compression of the spring. The potential energy stored in the spring (PEspring) when compressed is given by:
PEspring = 1/2 × k × x2
We equate the kinetic energy of the combined cars to the potential energy of the compressed spring to solve for x:
(1/2 × (M1 + M2) × vcombined2) = (1/2 × k × x2)
Solving for x gives us the maximum compression of the spring.