In the space station\'s cargo bay (assume Earthlike gravity), you\'re designing
ID: 1439140 • Letter: I
Question
In the space station's cargo bay (assume Earthlike gravity), you're designing a system to “catch” incoming cargo and slow it to a halt using springs. Freight containers weighing up to 15000 pounds will slide along a track until they are brought to rest by a combination of two coiled springs as illustrated in the figure. Both springs follow Hooke's law and have spring constants of 3200 N/m for the longer spring and 6800 N/m for the shorter spring. After the first spring compresses a distance of 3 m, the second spring acts together with the first to increase the force. If the cargo containers must come to rest 10 meters after first contacting the two-spring system, what is their maximum initial speed?
Note: Use work and energy calculations, not kinematics, to solve this problem. Look up Hooke's law for springs “in parallel” if you're not sure what the two springs pushing at once will do. You can use the figure above as the beginning of your diagram if you want.
Explanation / Answer
k1 = spring constant of longer spring = 3200
x1 = compression of longer spring = 10 m
k2 = spring constant of shorter spring = 3200
x2 = compression of shorter spring = 10 - 3 = 7 m
m = mass = 15000 pounds = 6803.88 kg
V = maximum speed
Using conservation of energy
maximum kinetic energy = total spring potential energy
(0.5) m V2 = (0.5) k1 x12 + (0.5) k2 x22
(6803.88) V2 = (3200) (10)2 + (6800) (7)2
V = 9.8 m/s