A bumper consisting of a nest of three springs is used to ar

A bumper, consisting of a nest of three springs, is used to arrest the horizontal motion of a large mass which is traveling at 32 ft/sec as it contacts the bumper. The two outer springs cause a deceleration proportional to the spring deformation. The center spring increases the deceleration rate when the compression exceeds 5 in. as shown on the graph. Determine the maximum compression x of the outer springs.

Solution

In this case for the outer springs deceleration is directly proportional to the spring deformation,

from the graph it is clear that for 5 inches of deformation deceleration is 1200 ft/sec²,

therefore constant of proportionality will be 240.

Total deceleration including combine effect of mid spring is from 1200-3600 ft/sec² , and the corresponding deformation is 5 inches, here mid spring increases deformation at double the rate i.e. 2400/5 = 480,

if we observe it, then deceleration caused by mid spring is same as two outer springs,

so the maximum compression of outer springs will be 10 inches,