A ball is thrown straight upward from the surface of the Moon. Is the maximum height it reaches less than, equal to, or greater than the maximum height reached by a ball thrown upward on Earth with the same initial speed? (Ignore air resistance in both cases.) Explain.

When analyzing projectile motion, it's essential to understand the conservation of energy. The principle states that energy cannot be created or destroyed, only transformed from one form to another. In the context of a ball thrown upward, the initial kinetic energy, given by the formula \( \frac{1}{2}mv^2 \), is gradually converted into gravitational potential energy (GPE), described as \( mgh \), where \( h \) is the height.
During its ascent, the projectile slows down because its kinetic energy decreases while its potential energy increases. At the peak of its trajectory, its velocity is zero, and all of its initial kinetic energy is now gravitational potential energy. Then, as the ball starts to fall, this potential energy is transformed back into kinetic energy. Because there is no external work done on the system and we ignore air resistance, the total mechanical energy of the ball remains constant throughout the projectile's flight. This interchange of energy forms is what we refer to as the conservation of energy in projectile motion.

Understanding acceleration due to gravity is crucial when exploring projectile motion. Acceleration due to gravity, symbolized as \( g \), is the constant rate at which objects accelerate when in free fall, given the absence of other forces such as air resistance.
On different celestial bodies, \( g \) varies due to differences in mass and radius. On Earth, \( g \) is approximately \( 9.8 \text{ m/s}^2 \). On the Moon, however, \( g \) is roughly \( 1.6 \text{ m/s}^2 \) because the Moon's mass and size are less than Earth's. This variation in \( g \) plays a pivotal role in determining the behavior of projectiles, such as the height they can reach. Given the same initial velocity, a projectile on the Moon would be less inhibited by the weaker lunar gravity and thus reach a greater height than on Earth.

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. The formula for calculating GPE is \( U = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is the height above the reference point.
In projectile motion, as a ball is thrown upwards, its gravitational potential energy increases, reaching a maximum at the highest point. This maximum GPE directly correlates with the maximum height that the ball can achieve. It's noteworthy that the weaker the gravitational field (or lower the value of \( g \) as on the Moon), the higher the ball can go with the same amount of initial kinetic energy. Therefore, gravitational potential energy is a useful concept for understanding how high a projectile can travel in different gravitational conditions.

The vertical motion equation for a projectile thrown upward can be expressed as \( h = \frac{v^2}{2g} \), where \( h \) is the maximum height, \( v \) is the initial velocity, and \( g \) is the acceleration due to gravity.
This equation is derived from the energy conservation principles and can be used to calculate the peak altitude of a projectile when its velocity reaches zero. It's particularly useful because it indicates the inverse relationship between the acceleration due to gravity and the height reached: as \( g \) decreases, \( h \) increases proportionally for a given initial velocity. Therefore, an understanding of this equation allows for predictions of a projectile's maximum height in different environments, such as comparing the height reached on Earth versus the Moon.