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Kinematics is a branch of physics that describes the motion of objects without considering the forces causing the motion. In the case of the rock being thrown upward, we can understand its trajectory through equations that relate velocity, acceleration, and time.
The initial velocity (\(v_0 = 18.0 \text{ m/s}\)) is the speed at which the rock is thrown. As it moves upward, gravity affects its motion by slowing it down until it stops for an instant at its peak height. Here, the final velocity momentarily becomes zero.
Using the kinematic equation \(v_t = v_0 - gt\), we determine how long it takes for the rock to reach this peak. This is crucial for understanding how long it stays airborne. After reaching the peak, gravity accelerates the rock downward, increasing its speed until it impacts the ground, leaving us with an initial upward trajectory followed by a downward descent.
So, the study of kinematics provides a means to predict how long the rock remains in the air, and its speed in various phases of its motion.

Energy conservation is a fundamental principle used to solve problems involving projectile motion, especially when no external work is done on the system, like in our rock-throwing scenario. This principle states that the total mechanical energy remains constant if only conservative forces, like gravity, are at play.
Initially, the rock possesses both kinetic energy \(\frac{1}{2}m v_0^2\) due to its motion and potential energy \(mgh\) due to its position above the ground. As the rock ascends, its kinetic energy decreases while potential energy increases until it reaches maximum height. At this point, potential energy is at its peak, and kinetic energy is minimal.When the rock descends, the energy conversion reverses, potential energy converts back into kinetic energy, thus increasing the speed of the rock as it falls. By applying the equation \(\frac{1}{2}m v_0^2 + mgh = \frac{1}{2}m v^2\), and solving for the final velocity, we find the speed of the rock just before hitting the ground is approximately 28.0 m/s. This illustrates how energy conservation helps quantify changes in velocity during projectile motion.

Free fall describes the motion of an object under the influence of gravity alone, with air resistance negligible. Once the rock reaches the peak of its flight, it falls under the exclusive influence of gravity. This phase is crucial as it determines how fast and how far the rock travels before hitting the ground.
During free fall, the rock accelerates downwards at \(g = 9.81 \text{ m/s}^2\), gaining speed as it moves. To calculate the time it takes to fall from its peak, we use the kinematic formula \(h = \frac{1}{2}gt^2\), where \(h = 46.5 \text{ m}\) is the total fall height. Solving for \(t\) gives us the time from the peak to the street.By understanding free fall dynamics, we determine not only how long the rock remains in the air but also how its speed and trajectory are affected by height and gravitational force. This comprehensive insight into free fall adds clarity to the predicted landing time and impact speed of any projectile moving under gravity.