A 8,000 -kg freight car rolls along rails with negligible friction. The car is b

Question

A 8,000-kg freight car rolls along rails with negligible friction. The car is brought to rest by a combination of two coiled springs as illustrated in the figure below. Both springs are described by Hooke's law and have spring constants k1 = 1,700 N/m and k2 = 3,200 N/m.After the first spring compresses a distance of x1 = 31.8 cm, the second spring acts with the first to increase the force as additional compression occurs as shown in the graph. The car comes to rest x2 = 49.5 cm after first contacting the two-spring system. Find the car's initial speed.

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A 8,000-kg freight car rolls along rails with negligible friction. The car is brought to rest by a combination of two coiled springs as illustrated in the figure below. Both springs are described by Hooke's law and have spring constants k1 = 1,700 N/m and k2 = 3,200 N/m. After the first spring compresses a distance of x1 = 31.8 cm, the second spring acts with the first to increase the force as additional compression occurs as shown in the graph. The car comes to rest x2 = 49.5 cm after first contacting the two-spring system. Find the car's initial speed.

Explanation / Answer

W1 = ( 1/2 ) k1 x12 = 0.5 (1 700 N/m) (0.495 m)2 = 208.27 J

W2 = ( 1/2 ) k2 x22 = 0.5 (3 200 N/m) (0.177 m)2 = 50.13 J

WTot = W1 + W2 = 200 J + 68 J = 258.40 J

KEi= ( 1/2 ) m vi2

KEi= ( 1/2 ) m vi2 = ( 1/2 ) (8 000 kg) vi2 = 258.40 J = Wsprings

vi2 = 0.0646 m2 / s2

vi = 0.254 m/s <--------- ANSWER

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