The force constant of the spring is approximately 650 N/m, and the force needed to compress the spring to the specified distance is approximately 87.75 N.
To calculate the force constant (k) of the spring in the BB gun, we use the conservation of energy, where the potential energy stored in the spring when it is compressed is equal to the kinetic energy of the plunger when it has reached its top speed. The potential energy in the spring is given by ½kx², and the kinetic energy of the plunger is given by ½mv², where x is the compression distance, m is the mass of the plunger, and v is the top speed. Setting these equal and solving for k gives us k = (mv²) / (x²). Given m = 0.0775 kg, v = 26.0 m/s, and x = 0.135 m, the calculated force constant is k ≈ 650 N/m, matching closest to option c).
To calculate the force exerted to compress the spring (F), we use Hooke's Law, F = kx. Using the previously determined value of k (650 N/m) and the compression distance x (0.135 m), we get F = 650 N/m * 0.135 m = 87.75 N, which is closest to option a).