Final answer:
To find the maximum height the marble can reach when compressed, we can equate the potential energy stored in the spring to the gravitational potential energy at the maximum height. For a compression of 2 cm, the marble can reach a maximum height of approximately 0.747 m, and for a compression of 4 cm, the marble can reach a maximum height of approximately 5.8 m.
Step-by-step explanation:
To find the maximum height the marble can reach, we can use the concept of mechanical energy conservation. When the spring is compressed, it contains potential energy which is transferred to the marble when it is launched. At the maximum height, all of the potential energy is converted to gravitational potential energy, given by the equation mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
(a) When the spring is compressed 2 cm (0.02 m), the potential energy is given by 1/2 kx^2, where k is the spring constant and x is the compression. So, the potential energy is 1/2(220 N/m)(0.02 m)^2 = 0.022 J. Equating this to mgh, we have 0.022 J = (0.003 kg)(9.8 m/s^2)h. Solving for h gives us h = 0.022 J / (0.003 kg)(9.8 m/s^2) ≈ 0.747 m.
(b) When the spring is compressed 4 cm (0.04 m), the potential energy is given by 1/2(220 N/m)(0.04 m)^2 = 0.176 J. Equating this to mgh, we have 0.176 J = (0.003 kg)(9.8 m/s^2)h. Solving for h gives us h = 0.176 J / (0.003 kg)(9.8 m/s^2) ≈ 5.8 m.