To solve this problem, we need to consider the conservation of mechanical energy. The initial mechanical energy of the system, which consists of the mass and the spring, is equal to the final mechanical energy when the spring is compressed.
The initial mechanical energy of the system includes both the kinetic energy and the potential energy due to the spring's compression:
Initial mechanical energy = (1/2)mv0^2 + (1/2)kx^2
Where: m = mass of the object (0.05 kg) v0 = initial velocity (5 m/s) k = spring constant (10 N/m) x = compression of the spring (0.05 m)
Since the spring is compressed, the potential energy term becomes negative:
Initial mechanical energy = (1/2)mv0^2 - (1/2)kx^2
The final mechanical energy is entirely kinetic energy since the spring is compressed and there is no potential energy:
Final mechanical energy = (1/2)mv^2
We can equate the initial and final mechanical energies and solve for the final velocity (v):
(1/2)mv0^2 - (1/2)kx^2 = (1/2)mv^2
(1/2)(0.05 kg)(5 m/s)^2 - (1/2)(10 N/m)(0.05 m)^2 = (1/2)(0.05 kg)v^2
0.625 J - 0.00125 J = 0.025 kg v^2
0.62375 J = 0.025 kg v^2
Divide both sides by 0.025 kg:
v^2 = 0.62375 J / 0.025 kg
v^2 = 24.95 m^2/s^2
Take the square root of both sides to find the final velocity (v):
v = √(24.95 m^2/s^2)
v ≈ 4.995 m/s
Therefore, when the spring is compressed 5 cm, the mass's velocity will be approximately 4.995 m/s.