Final answer:
The force constant of the spring is calculated using the energy conservation principle, resulting in an unexpected value of 888.89 N/m. The force needed to compress the spring is calculated using Hooke's law, yielding 133.33 N. Neither matched the given multiple choice answers, indicating a discrepancy in the question or calculations.
Step-by-step explanation:
To calculate the force constant of the plunger's spring, we'll use the conservation of energy principle which states that the potential energy stored in the compressed spring is equal to the kinetic energy of the moving plunger. The equation for the potential energy (PE) of a compressed spring is PE = (1/2)kx2 and the kinetic energy (KE) of the plunger is KE = (1/2)mv2.
Setting the potential energy equal to the kinetic energy gives us:
(1/2)k(0.150 m)2 = (1/2)(0.0500 kg)(20.0 m/s)2
After solving for the spring constant k, we have:
k = (0.0500 kg)(20.0 m/s)2 / (0.150 m)2
k = (0.0500)(400) / (0.0225)
k = 20 / 0.0225
k = 888.89 N/m
The calculated value is not on the list of multiple choice options, indicating a potential error in the given choices or the calculations. Professional integrity requires us to indicate this discrepancy.
To calculate the force needed to compress the spring, we use Hooke's law, which is F = kx.
F = (888.89 N/m)(0.150 m)
F = 133.33 N
Again, the calculated force is not among the multiple choice answers, underscoring the discrepancy.