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The concept of free-fall distance is fundamental in various physics problems. Free-fall describes the motion of an object under the influence of gravitational force only. When ignoring air resistance, an object in free-fall accelerates downward at a constant rate. This rate is known as the acceleration due to gravity, denoted by the symbol \( g \), which is approximately \( 9.8 \, m/s^2 \) on Earth.
To calculate the distance an object travels during free-fall, we use the formula: \[ h = \frac{1}{2} g t^2 \] where:

• \( h \) is the height or distance fallen (in meters).
• \( g \) is the acceleration due to gravity (\( 9.8 \, m/s^2 \)).
• \( t \) is the time taken (in seconds).

The formula shows that the distance fallen is proportional to the square of the time the object has been falling. This quadratic relationship means that as the object falls for a longer period, the distance it covers increases rapidly.

For example, in the exercise given, the diver starts from a height of \( 10 \) meters with an initial vertical velocity of zero. By plugging these values into the formula, we determine the time it takes for the diver to reach the water by solving for \( t \). This step is crucial as it sets the stage for calculating other parameters like rotational velocity.

Angular displacement is an important concept in rotational motion. It refers to the angle through which an object has rotated or moved in a given period. It is typically measured in radians but can also be measured in degrees.
For any object making full rotations, one complete revolution is equivalent to \( 2\pi \) radians. Hence, if an object makes multiple revolutions, its total angular displacement is the number of revolutions multiplied by \( 2\pi \).

• Angular displacement \( \theta \) can be calculated using the formula: \[ \theta = n \times 2\pi \] where \( n \) is the number of revolutions.

In the exercise, the diver makes \( 2.5 \) revolutions during the dive. The angular displacement \( \theta \) thus becomes: \[ \theta = 2.5 \times 2\pi \] This results in an angular displacement of \( 5\pi \) radians. Understanding this concept is essential for solving problems related to rotational motion, as it directly impacts the calculation of quantities like rotational velocity.

Rotational velocity refers to how fast an object rotates or turns around an axis. Understanding this concept is crucial for problems involving objects that spin or rotate. The average rotational velocity is given by the formula: \[ \omega_{avg} = \frac{\theta}{t} \] where:

• \( \omega_{avg} \) is the average rotational velocity (in radians per second).
• \( \theta \) is the angular displacement (in radians).
• \( t \) is the time taken (in seconds).

In the given problem, we already have the angular displacement \( \theta = 2.5 \times 2\pi \) radians and the time \( t \approx 1.43 \ s \). Plugging these values into the formula gives us the average rotational velocity:
\[ \omega_{avg} = \frac{2.5 \times 2\pi}{1.43} \approx 10.99 \, rad/s \] This result shows how quickly the diver is spinning as they fall. Understanding the rotational velocity formula and how to apply it is essential for analyzing motion, whether it's a spinning wheel, a rotating planet, or, in this case, a diver performing revolutions during a fall.