Final answer:
The wrecking ball needs sufficient height to convert potential energy into kinetic energy to reach the target velocity. Using the conservation of energy, the calculated height exceeds the provided options, indicating a discrepancy in the question or options. None of the suggested heights would allow the ball to achieve the required velocity.
Step-by-step explanation:
The question involves applying the principles of conservation of energy to calculate the height from which an 800 kg wrecking ball must be dropped to achieve a velocity of 50 m/s at the bottom of its swing. When the ball is at rest at the top of the swing, its potential energy (PE) will be equal to the kinetic energy (KE) it has at the bottom of the swing. The potential energy at the top can be calculated using PE = mgh, where m is the mass, g is the acceleration due to gravity (9.81 m/s2), and h is the height. The kinetic energy at the bottom is given by KE = 0.5mv2, where v is the velocity.
To solve for h, we set PE equal to KE giving us mgh = 0.5mv2. We can cancel the mass (m) from both sides since it doesn't change, thus giving us gh = 0.5v2. Substituting in the values for g and v, and solving for h, will give us the required height.
When substituting in g = 9.81 m/s2 and v = 50 m/s, we get 9.81h = 0.5 × (50)2, which simplifies to h = (0.5 × 2500) / 9.81 ≈ 127.42 m. Since 127.42 m is not one of the provided options (A) 10 m, (B) 20 m, (C) 30 m, (D) 40 m, there seems to be a discrepancy. Assuming the options are correct, none of them would be able to provide the required velocity of 50 m/s as they are all too low.