Final answer:
The work done by the spring during recoil is calculated as the change in potential energy between the initial and final displacements. The spring does positive work amounting to 43.515 J, releasing energy as it recoils from 3.4 m to 1.9 m with a spring constant of 11 N/m.
Step-by-step explanation:
The work done by a spring force is calculated as the change in potential energy, which is given by the equation U = ½ kx², where k is the spring constant and x is the displacement from equilibrium. In this scenario, as the spring recoils from 3.4 m to 1.9 m, the work done is the difference in potential energy at these displacements. Using the given spring constant (k = 11 N/m), the work done by the spring during recoil can be calculated as:
- Initial potential energy at 3.4 m: U1 = ½ * 11 N/m * (3.4 m)²
- Final potential energy at 1.9 m: U2 = ½ * 11 N/m * (1.9 m)²
The work done by the spring is U1 - U2. We can calculate the values:
- U1 = 0.5 * 11 * 3.4² = 62.38 J
- U2 = 0.5 * 11 * 1.9² = 18.865 J
Therefore, the work done as the spring recoils is 62.38 J - 18.865 J = 43.515 J. Because the final potential energy is lower, the spring has done positive work on the system, converting potential energy into kinetic energy or transferring it to the environment.