Final answer:
To find the minimum initial speed needed to throw a baseball to reach 15.3 m, we use the conservation of energy principle to equate kinetic energy at launch to the potential energy at the peak height, then solve for the initial velocity.
Step-by-step explanation:
To calculate the minimum initial speed needed for a 175 g baseball to reach a height of 15.3 m from an initial height of 1.56 m using conservation of energy, we need to equate the gravitational potential energy at the highest point to the kinetic energy at the starting point, as the system is closed and energy is conserved.
First, we express the baseball's mass in kilograms: 175 g = 0.175 kg. The gain in gravitational potential energy (PE) can be calculated using the formula PE = m * g * h, where m is the mass, g is the acceleration due to gravity (9.8 m/s²), and h is the change in height. In this case, h is 15.3 m - 1.56 m = 13.74 m.
The potential energy at the highest point is then PE = 0.175 kg * 9.8 m/s² * 13.74 m
To find the initial kinetic energy (KE), we solve for KE using the formula KE = ½ * m * v^2, where m is the mass and v is the velocity. Setting the KE equal to the PE gives us ½ * 0.175 kg * v^2 = PE.
Solving for v gives us the square root of (2 * PE / m). Substituting the PE we found earlier and simplifying provides the minimum initial speed the baseball must have.
Through this process, we apply principles from mechanics, specifically the work-energy theorem which underpins the conservation of mechanical energy in physics.