91Ó°ÊÓ

Potential energy is a key concept in physics, representing the energy stored in an object due to its position, condition, or arrangement. In the context of our exercise, it is the energy associated with the object's height from the ground.

The formula used to calculate potential energy is:

• \( PE = mgh \)

Here, \( m \) is the mass of the object (in kilograms), \( g \) is the acceleration due to gravity (approximately 9.81 m/s² on Earth), and \( h \) is the height in meters above a reference point.

When the ball is at the top of the 6.10 m building, it has a maximum potential energy. As it falls, this potential energy is converted into kinetic energy up until the moment it hits the ground. Understanding potential energy helps us analyze how energy transforms as the ball moves through its trajectory.

Kinetic energy is the energy an object possesses due to its motion. As the ball falls from the building, its potential energy is transformed into kinetic energy. The formula for kinetic energy is:

• \( KE = \frac{1}{2}mv^2 \)

Here, \( v \) is the velocity of the object as it moves. However, in our exercise, an easier approach is to consider that the initial kinetic energy just before impact is equal to the initial potential energy at the top of the building: \( KE_0 = mgh \).

After each bounce, the ball loses 10% of its kinetic energy. This step-by-step decay of kinetic energy is important in determining how high the ball can bounce back. With each bounce, a lesser amount of kinetic energy means the ball reaches a lower height, until a point it cannot reach the 2.44 m windowsill.

An inelastic collision is one where some of the object's kinetic energy is not conserved and is transformed into other forms of energy, like heat or sound, during the collision. In our roadmap, after every collision with the ground, the ball loses a portion of its kinetic energy and hence bounces back to a lower height.

In this case, it's stated that 10% of the kinetic energy is lost with every bounce. This is mathematically represented by multiplying the kinetic energy after each bounce by 0.9: \( KE_n = 0.9 \cdot KE_{n-1} \).

Understanding inelastic collisions is crucial when evaluating how many times the ball can bounce and still reach a particular height, such as the windowsill in the exercise.

Motion dynamics involves studying the forces and energies involved in the movement of objects. This exercise primarily focuses on the transformation of energy forms as the ball moves under gravity, illustrating core principles of motion dynamics.

Here, the main forces acting are gravity, promoting the ball's movement downward and affecting its motion upon bouncing back. The interaction of these forces is quantified through the transition of potential energy into kinetic energy and the influence of inelastic collisions reducing the kinetic energy after each bounce.

By understanding motion dynamics, we can predict how far and how fast the ball moves, and with how much energy, in each part of its trajectory. Analyzing these factors is essential for solving how many bounces it takes for the ball to still meet the windowsill's height.