91Ó°ÊÓ

Kinematics is a branch of physics that focuses on the motion of objects without considering the forces that cause the motion. In projectile motion, kinematics plays a crucial role. We analyze how the rock moves both horizontally and vertically across the path of its trajectory.

Key kinematic equations help us break down the motion into easily understandable parts. These equations relate quantities like displacement, velocity, acceleration, and time. By using these formulas, we can determine how long it takes an object to move from one point to another and what speed is necessary to complete this motion.

In this exercise, kinematic equations allowed us to find the time the rock takes to fall and the initial speed needed to just clear the dam.

Vertical motion concerns how an object moves up or down, influenced by gravity. In this case, the rock's vertical movement starts as it rolls off a cliff.

Using the kinematic equation for free fall, we explored vertical motion to find how long it would take the rock to drop 20 meters. The equation is given by: \( h = \frac{1}{2}gt^2 \), where \( h \) is the height, and \( g \) is the acceleration due to gravity, 9.8 m/s\(^2\). By rearranging this equation, we solved for the time \( t \): \[ t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 20}{9.8}} \approx 2.02 \text{ seconds}.\]
This time indicates how long it takes the rock to hit the water level from the cliff.

While vertical motion dictates how fast the object falls, horizontal motion shows how far it travels. Horizontal velocity remains constant in the absence of air resistance.

Using the horizontal motion equation \( d = v_{x} \times t \), where \( d \) is the distance and \( v_{x} \) is the horizontal velocity, we find the minimum speed necessary to avoid the dam. By solving: \[ v_{x} = \frac{d}{t} = \frac{100}{2.02} \approx 49.5 \text{ m/s}. \]
This speed ensures that the rock has the ability to travel the 100 meters needed to clear the dam. The horizontal motion remains unaffected by gravity, so this speed sustains until it hits the ground.

Free fall describes any motion of an object where gravity is the only acting force. As soon as the rock leaves the cliff, it is in free fall.

The impact of free fall becomes clear when considering the total fall from the cliff to the plain. The additional 25 meters down from the dam level to the plain increases the total drop to 45 meters. This affects the fall time:\[ t_{total} = \sqrt{\frac{2 \cdot 45}{9.8}} \approx 3.03 \text{ seconds}. \]
The rock's total trajectory then allows us to calculate where it will impact the plain. Since it's in horizontal motion during this free fall, the total distance covered horizontally is \[ v_{x} \times t_{total} = 49.5 \times 3.03 \approx 150.0 \text{ m}. \]
Subtract the 100 meters needed to clear the dam, and you see the rock lands 50 meters beyond it.