To solve this problem, we can use the principle of conservation of mechanical energy. The mechanical energy of the ball at the maximum height is equal to the mechanical energy of the spring when it is released.
Let's calculate the spring launch speed of the ball first.
(a) To find the speed at which the spring launches the ball, we need to calculate the potential energy and the kinetic energy of the ball at the maximum height.
The potential energy of the ball at the maximum height can be calculated using the equation:
PE = m * g * h
where PE is the potential energy, m is the mass of the ball, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the maximum height.
PE = 0.059 kg * 9.8 m/s² * 1.74 m
PE = 0.098 J
The kinetic energy of the ball when it is launched can be calculated using the equation:
KE = 1/2 * m * v²
where KE is the kinetic energy, m is the mass of the ball, and v is the launch velocity.
We can equate the potential energy and the kinetic energy to find the launch velocity:
0.098 J = 1/2 * 0.059 kg * v²
Rearranging the equation, we get:
v² = (2 * 0.098 J) / 0.059 kg
v² = 3.322 m²/s²
Taking the square root of both sides, we find:
v = √(3.322 m²/s²)
v ≈ 1.82 m/s
Therefore, the spring launches the ball at a speed of approximately 1.82 m/s.
(b) To find the spring's initial compression distance, we can use the equation for the potential energy stored in the spring:
PE = 1/2 * k * x²
where PE is the potential energy, k is the spring constant, and x is the compression distance.
We can rearrange the equation to solve for the compression distance:
x = √(2 * PE / k)
Substituting the values into the equation, we get:
x = √(2 * 0.098 J / 780 N/m)
x ≈ 0.080 m
Therefore, the spring's initial compression distance is approximately 0.080 m.