91Ó°ÊÓ

In the context of projectile motion, free fall is a critical concept as it describes the motion of an object under the influence of gravitational force only. When an object is dropped from a certain height, it accelerates towards the earth due to gravity. In our exercise, the ball falls freely from a height of 45 meters until it reaches 25 meters, where it explodes.

The formula for distance traveled under constant acceleration, like gravity, is given by \[h = \frac{1}{2}gt^2\]where:

• \( h \) is the height or distance
• \( g \) is the acceleration due to gravity (\(9.8 \mathrm{~m/s}^2\))
• \( t \) is the time taken

Using this relationship, we calculated that it took about 2.02 seconds for the ball to fall from 45 meters to 25 meters. Free fall is solely influenced by gravity, which affects everything equally regardless of mass, making it a straightforward yet powerful concept in understanding motion.

Conservation of momentum is a fundamental principle in physics that describes how the momentum of a system remains constant when no external forces act upon it. In our scenario, when the ball explodes at a height of 25 meters, the momentum before the explosion and after must be equal.

Before the explosion, the ball had no horizontal velocity, implying zero horizontal momentum. After the explosion, though one piece moves horizontally at \(10 \, \mathrm{m/s}\), the law of conservation of momentum dictates that the second piece must move in the opposite direction at the same speed to keep the total horizontal momentum zero: \[m \times 10 + m \times v = 0\]where \( v \) is the velocity of the second piece, resulting in \( v = -10 \, \mathrm{m/s}\).

This concept showcases the balance and symmetry that physics often maintains within isolated systems, ensuring that any action has an equal and opposite reaction, in terms of momentum in this case.

Two-dimensional motion occurs when an object moves in a plane, having both vertical and horizontal components, such as in the case of projectile motion. In the exercise, after the explosion, one piece of the ball moves horizontally, thus illustrating two-dimensional motion.

The horizontal motion is simple; it moves at a constant speed as no external forces act horizontally. The formula \[d = v \times t\]calculates the distance traveled:

• \(d\) is the distance
• \(v\) is the velocity (\(10 \, \mathrm{m/s}\) for one piece)
• \(t\) is the time (\(2.26 \, \mathrm{s}\) till it hits the ground)

Thus, it moves \(22.6 \, \mathrm{m}\) horizontally before striking the ground. The total movement comprises both horizontal and vertical trajectories, showcasing a comprehensive real-world application of physics.

Breaking down the motion into its components simplifies complex interactions and aligns theoretical predictions with practical observations.