Final answer:
The force constant of a spring can be calculated by using the conservation of energy principle, where the initial kinetic energy of the moving box is converted into the elastic potential energy of the spring at maximum compression.
Step-by-step explanation:
To determine the force constant of the spring (often represented by k), we can use the conservation of energy principle. Since the surface is frictionless, mechanical energy is conserved, that is, the initial kinetic energy (KE) of the box equals the elastic potential energy (PEspring) stored in the spring at maximum compression. The initial KE of the box is calculated using KE = 0.5 * mass * velocity2. With a mass of 3.1 kg and an initial speed of 1.8 m/s, the KE becomes 0.5 * 3.1 kg * (1.8 m/s)2.
The elastic potential energy stored in the spring at maximum compression is given by PEspring = 0.5 * spring constant * compression2. To come to rest, the box must have compressed the spring by 5.3 cm, which we convert to meters (0.053 m) for consistency in units. Equating the kinetic energy to the potential energy of the spring, we get 0.5 * 3.1 kg * (1.8 m/s)2 = 0.5 * k * (0.053 m)2. Solving for k, we find the spring constant. Finally, plug in the values to solve for k, which will give you the force constant of the spring in Newtons per meter (N/m).