Final answer:
To determine the initial compression of the spring, we can use the principle of conservation of mechanical energy. The potential energy stored in the compressed spring is equal to the potential energy gained by the ball when it reaches the maximum height. Given the mass of the ball and the height it reaches, we can calculate the initial compression of the spring to be 450 mm.
Step-by-step explanation:
To determine the initial compression of the spring, we can use the principle of conservation of mechanical energy. The potential energy stored in the compressed spring is equal to the potential energy gained by the ball when it reaches the maximum height. The formula to calculate the potential energy in a spring is:
PE = 0.5 * k * x^2
where PE is the potential energy, k is the spring constant, and x is the compression or extension of the spring. Rearranging the formula, we have:
x = sqrt(2 * PE / k)
Given that the mass of the ball is 312 grams and the height it reaches is 10 meters, we can calculate the potential energy:
PE = m * g * h
where m is the mass, g is the acceleration due to gravity, and h is the height. Substituting the values, we get:
PE = 0.312 kg * 9.8 m/s^2 * 10 m = 30.576 J
Now, plugging the values into the formula for the compression, we get:
x = sqrt(2 * 30.576 J / 44 N/m) = 0.450 m = 450 mm
Therefore, the initial compression of the spring is 450 mm.