Final answer:
Using the conservation of energy, the spring's potential energy when compressed is converted to the ball's kinetic energy. By calculating the potential energy stored in the spring and equating it to kinetic energy, the maximum speed of a 500 g ball released from a spring compressed by 0.40 m can be determined to be approximately 12.5 m/s.
Step-by-step explanation:
To find out the maximum speed that a spring with a spring constant of 850 N/m can give to a 500 g ball when it is compressed 0.40 m, we use the principle of conservation of energy. The potential energy stored in the compressed spring is fully converted into the kinetic energy of the ball at the moment the spring releases its compression and the ball leaves the spring. We formulate this using Hooke's Law for spring potential energy and the kinetic energy expression.
The potential energy (PE) stored in the compressed spring is given by:
PE = 1/2 k x2, where k is the spring constant and x is the displacement from its equilibrium position.
PE = 1/2 (850 N/m) (0.40 m)2 = 68 J
This energy is converted into kinetic energy (KE) which is given by:
KE = 1/2 m v2, where m is the mass and v is the final speed of the ball.
By setting the potential energy equal to the kinetic energy: 1/2 k x2 = 1/2 m v2, we can solve for v for maximum speed of the 500 g ball.
68 J = 1/2 (0.5 kg) v2
v = sqrt(2 * 68 J / 0.5 kg)
v ≈ 13.07 m/s
Therefore, the closest answer choice to the calculated maximum speed is 12.5 m/s.