91Ó°ÊÓ

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. It focuses on parameters such as displacement, velocity, acceleration, and time.
This exercise involves analyzing the motion of divers who jump off a platform, using the basic kinematic equation for free fall.
In kinematics, especially when analyzing vertical motion, the free fall equation for final velocity is often employed:

• \(v = \sqrt{2gh}\)

This formula calculates the final speed of an object (such as a diver) when it is dropped from a certain height \(h\), given that the acceleration due to gravity \(g\) is 9.8 m/s². It is essential for understanding how objects accelerate when falling freely under gravity's influence.

The principle of energy conservation states that energy cannot be created or destroyed in an isolated system. Instead, it can only be transformed from one form to another. In the context of this exercise, we are dealing with kinetic and potential energy during the diver's ascent and descent.

• Initially, at the board level, divers have potential energy due to their height: \(P.E_{initial} = mgh\).
• When a diver leaps up, they convert some of this potential energy to kinetic energy: \(K.E_{initial} = \frac{1}{2}mv_i^2\).
• Upon entering the water, the total energy is kinetic: \(K.E_{final} = \frac{1}{2}m(25)^2\).

By applying energy conservation, the problem calculates if a leap can result in a 25 m/s entry speed, solving for the necessary initial velocity of the leap.

Free fall refers to the condition of an object accelerating under the influence of gravity alone, without any resistance, such as air friction. In this situation, the divers step off a platform and accelerate toward the water with an acceleration of 9.8 m/s².
It's important to note:

• The divers' motion is strictly governed by gravitational pull.
• Utilizing the free fall kinematics formula, the exercise demonstrates that the ending velocity due to this fall is 20.47 m/s from a height of 21.3 meters.
• Since air resistance is ignored, this simplification allows us to calculate the velocity using straightforward equations.

Understanding free fall helps explain why the announcer's claim of a 25 m/s free-fall entry velocity is incorrect.

Velocity calculations in physics require understanding both magnitude and direction of an object's movement. To calculate the final velocity after free-fall, the formula \(v = \sqrt{2gh}\) is used, resulting in a value of approximately 20.47 m/s.
Furthermore, if the diver leaps upward, the velocity on entering the water must be recalculated by considering both the initially supplied upward speed and gravitational deceleration.
In this case:

• The formula was rearranged to determine the initial velocity needed for divers to reach 25 m/s downhill: \[v_i = \sqrt{25^2 - 2 \times 9.8 \times 21.3}\] yielding \(v_i \approx 9.18 \, m/s\).
• Such calculations need to incorporate both the increase from the initial leap and the subsequent increase due to free fall from height.

These calculations explain why achieving the stated condition by human leap alone is impractical.