91Ó°ÊÓ

Free fall describes the motion of a body where gravity is the only force acting upon it. When objects are dropped, they accelerate towards the Earth without any initial push or force other than gravity. Gravity on Earth provides an acceleration of approximately \(9.8 \, \text{m/s}^2\).

This means that for every second an object is in free fall, its speed increases by 9.8 meters per second. During free fall, objects do not encounter air resistance or any other forces. This makes calculations simpler since only the constant acceleration due to gravity is involved.

• Initial velocity for dropped objects is zero.
• Gravity provides constant acceleration.
• Air resistance is typically ignored in basic free fall scenarios.

The final velocity of an object in free fall is the speed it reaches right before it hits the ground. Starting from rest, the final velocity can be calculated using the equation:
\[ v = v_0 + gt \]
where \(v\) is the final velocity, \(v_0\) is the initial velocity, \(g\) is the acceleration due to gravity, and \(t\) is the time the object spends falling.

In the exercise, the marble's initial velocity \(v_0\) is zero because it is dropped. With \(g = 9.8 \, \text{m/s}^2\) and \(t = 5 \, \text{s}\), the final velocity is:
\[ v = 0 + 9.8 \times 5 = 49 \, \text{m/s} \]
Thus, the marble strikes the water at 49 meters per second.

Final velocity provides insight into how fast an object is moving just before impact, a crucial component in understanding motion.

Calculating the height from which an object falls involves understanding how distance is covered during free fall. The height or distance, \(h\), can be determined with the equation:
\[ h = v_0t + \frac{1}{2}gt^2 \]
In scenarios like our exercise, since the object is dropped, \(v_0\) is zero. This simplifies the equation to only account for the distance covered due to gravity's acceleration:
\[ h = \frac{1}{2} \cdot 9.8 \cdot 5^2 \]
Calculating gives us:
\[ h = \frac{1}{2} \cdot 9.8 \cdot 25 = 122.5 \text{ meters} \]
Thus, the bridge is 122.5 meters high.

Knowing the height helps visualize and measure the extent of the motion, a key element in physics studies.

Equations of motion are essential tools in kinematics. They allow us to predict and describe the motion of objects. For free fall scenarios, there are three primary equations to use:

• \( v = v_0 + gt \) for calculating final velocity.
• \( h = v_0t + \frac{1}{2}gt^2 \) for determining height.
• \( v^2 = v_0^2 + 2gh \) which can be used to find any of the variables if others are known.

These equations are based on the principles of constant acceleration. When applying them, it’s crucial that assumptions like negligible air resistance hold true.

By understanding these equations, we can solve a wide range of problems involving motion, whether it is a marble from our example or any other object moving under gravity's influence.