Final answer:
To find how high the marble will go and its velocity when it leaves the gun, we use the stored potential energy in the compressed spring. This energy is converted to kinetic energy and then to gravitational potential energy, allowing us to calculate the maximum height and release velocity of the marble.
Step-by-step explanation:
To determine a) how high the marble will go and b) how fast it will be going when it leaves the gun, we can use the principles of energy conservation and mechanics. The potential energy stored in the compressed spring will convert into kinetic energy at the point of release and then into gravitational potential energy at the maximum height.
Firstly, the potential energy stored in the spring (Us) when compressed is given by the formula Us = 1/2 k x^2, where k is the spring constant and x is the compression distance. Plugging in the values, we get Us = 1/2 (120 N/m) (0.04 m)^2 = 0.096 J. This energy will be entirely converted into gravitational potential energy (Ug) at the maximum height. The gravitational potential energy is given by Ug = mgh, where m is the mass of the marble, g is the acceleration due to gravity (9.8 m/s^2), and h is the maximum height. Equating Us and Ug and solving for h, we find the maximum height.
The velocity of the marble when it leaves the gun can be found using the kinetic energy formula Ke = 1/2 mv^2, where m is the mass and v is the velocity. The kinetic energy at the point of release is equal to the potential energy stored in the spring, so Ke = Us. Solving for v will yield the velocity at which the marble leaves the spring gun.