Final answer:
The work done by gravity on the lead ball by the time it reaches its maximum height is -18.75 J because the force of gravity does negative work by reducing the ball's kinetic energy, equating to option b. The correct multiple-choice option is (b).
Step-by-step explanation:
To determine the work done by the force of gravity on the lead ball by the time it reaches its maximum height, you have to consider the definition of work in physics, which is the product of the displacement and the component of the force along the displacement. Since gravity only does work during the vertical movement of the ball and the displacement is in the same direction as the gravitational force (but opposite in direction), we can calculate work with the equation W = -mgh, where m is the mass of the ball, g is the acceleration due to gravity (9.8 m/s²), and h is the height the ball reaches.
The initial kinetic energy (KE) of the ball can be used to find the maximum height since KE at launching will be completely converted to gravitational potential energy (PE) at the maximum height, with PE = mgh. From the initial velocity (v) we have KE = 0.5mv², and setting KE equal to PE we get 0.5mv² = mgh. Solving for h gives us h = (0.5 × 0.125 kg × 30 m/s²) / (0.125 kg × 9.8 m/s²), which simplifies to h = (0.5 × 30² / 9.8).
After calculating h, we can find the work done by gravity as W = -mgh. However, since we are interested in the work done by gravity before it reaches the maximum height, we recognize that it has done negative work by taking away the ball's kinetic energy. Therefore, the work done by gravity will be equal to the negative of the potential energy the lead ball has at its peak height, which is the initial kinetic energy with a negative sign. This work can be expressed as -0.5mv², which equals -18.75 J (option b). So, before the ball reaches the maximum height, the work done by gravity is -18.75 J. The correct multiple-choice option is (b).