91Ó°ÊÓ

Understanding the equations of motion is crucial to solving problems involving freely falling objects. These equations describe how the velocity of an object changes under the influence of a constant acceleration, such as gravity. In physics, the most commonly used equations for objects moving under uniform acceleration are:

• First equation: \( v = u + at \) (relates velocity, initial velocity, and acceleration)
• Second equation: \( s = ut + \frac{1}{2}at^2 \) (relates distance, initial velocity, and acceleration)
• Third equation: \( v^2 = u^2 + 2as \) (relates final velocity, initial velocity, and distance)

In these equations, \(s\) is the displacement, \(u\) is the initial velocity, \(v\) is the final velocity, \(a\) is the acceleration, and \(t\) is the time. For freely falling objects, the acceleration \(a\) is equal to the acceleration due to gravity, denoted as \(g\), and it is a constant value of approximately \(9.8 \text{ m/s}^2\) on Earth.

The concept of initial velocity, denoted as \(u\), is a fundamental aspect of kinematics and refers to the velocity of an object before it has been influenced by an external force or acceleration. In the case of our freely falling object, the initial velocity is the speed at which the object was traveling before it entered the final segment of its fall, measured over the last \(30.0 \text{ m}\). Knowing the initial velocity is critical for calculating the total time taken for an object to reach the ground from a certain height. It is the starting point for using the equations of motion to solve for other unknown variables such as total displacement and time.

The total distance fallen by an object, often represented as \(s\), is the entire vertical distance that it travels during its fall. It is a straightforward concept but requires careful calculation, as the object might have had an initial velocity other than zero when it began its descent. In order to determine the total height from which the object has fallen, we must account not only for the distance covered during observation (for example, the last \(30.0 \text{ m}\) mentioned in the question) but also for the distance fallen prior to this period. This can be obtained by summing up the initial velocity-induced displacement and the displacement caused by the acceleration due to gravity over the total time.

The acceleration due to gravity, denoted as \(g\), is a constant that determines the acceleration experienced by objects when they fall toward Earth without any resistance from the air. It is approximately \(9.8 \text{ m/s}^2\) at the Earth's surface. The effect of gravity is the same on all objects, regardless of their mass, resulting in all free-falling objects accelerating at the same rate when air resistance is negligible. This concept is pivotal when solving for the total distance an object has fallen, as it helps us understand how quickly the velocity of the object increases as it continues to fall.