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# Tugboat, Conservation of energy question?

A tugboat T having a mass of 19,000kg is tied to a barge B having a mass of 75000 kg . If the rope is elastic such that it has a stiffness k = 600 kN/m, determine the maximum stretch in the rope during the initial towing. Originally both the tugboat and the barge are moving in the same direction, the tugboat has a speed of 4.1666 m/s and the barge has a speed of 2.7777 m/s. Neglect the resistance of the water.

The question is a little vague but i assume that the tugboat is moving at a constant speed and the tension in the rope is zero once they are moving at the same speed as there's no water friction. The maximum stretch is 0.23m how do I get this?

### 2 Answers

- oldschoolLv 79 years agoFavorite Answer
What will happen is the tug will slow down and the barge will speed up.

The momentum before will equal the momentum after

75k*2.7777+19k*4.1666 = [19k+75k]*V = V = 3.06m/s

The change in kinetic energy for the barge = 1/2 *75k*[3.06^2-2.7777^2] = 61,440J

The change in kinetic energy for the tug = 1/2 *19k*[4.1666^2-3.06^2] = 76,062J

14,622J is the difference

That energy stretched the rope over a distance d and is therefore stored in that

stretched rope temporarily since water resistance is neglected:

1/2 *k*d^2 = 14,622 J =1/2 * 600k * d^2 => d = 0.22m

Answer slightly different must be due to round off