A freight car of mass 1800kg is timed at 4.2m/s just after it runs into a spring

Question

A freight car of mass 1800kg is timed at 4.2m/s just after it runs into a spring-loaded bumper at the end of the track. The bumper consists of an 800kg mass that the car runs into, and a pair of large springs. The car travels 2.4m before coming to rest.

a)using an appropriate conservation law, find the speed of the freight car before it struck the bumper. b)Find the time taken for the car to be brought to rest.

Note: Please do not copy paste the answer archived on Chegg. That answer is not correct.

Explanation / Answer

a) So all the energy should be with the freight car
KE = 0.5 (1800 + 800) (4.2)^2 = 0.5 1800 v^2
v = sqrt ( 2600/1800) * 4.2 = 5.05 m/s

b) you need to find k
P.E. spring = 0.5 k x^2 = 0.5 k (2.4)^2 = K.E. = 0.5 2600 (4.2)^2
k = 7962.5 N/m

F = ma = kx
a = kx/m = k/m x
a = x'' = k/m x
Uh, it's been a while for differential equations, so if you how to solve for x(t) then you can solve for x(t) = -2.4. Your initial conditions are that x(0) = 0 and v(0) = -4.2
I think x(t) = e^[sqrt(k/m) * t] - [4.2 + sqrt(k/m)] t - 1

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