91Ó°ÊÓ

Kinetic energy is the energy an object has due to its motion. Any object that moves possesses kinetic energy. In our problem, an object initially at rest breaks into two pieces due to an internal explosion. This explosion distributes the total energy released as kinetic energy to the two separate pieces. The key formula for kinetic energy (KE) is:

• \( KE = \frac{1}{2}mv^2 \)

where \(m\) is the mass of the object and \(v\) is its velocity.
This formula tells us that kinetic energy depends on both mass and speed—more mass or greater speed means more kinetic energy.
In our exercise, the total kinetic energy from the explosion is known to be 7500 J. This energy is then split between the two pieces of the object. Given that one piece has 1.5 times the mass of the other, the challenge is to determine how this energy is shared between the two pieces based on their masses and velocities.

The principle of energy conservation states that in a closed system, energy cannot be created or destroyed; it only changes form. This is a fundamental concept in physics. In the context of our problem, energy conservation tells us that the total energy before and after the explosion remains consistent.
Before the explosion, the object is at rest, so its kinetic energy is 0. During the explosion, 7500 J of internal energy is converted into kinetic energy that propels the pieces apart.

• The total kinetic energy after the explosion is the same as the total energy released, 7500 J.
• All of this energy becomes kinetic energy for the two moving pieces: \( KE_1 + KE_2 = 7500 \, \mathrm{J} \).

By applying energy conservation, we calculated the kinetic energies of the pieces to be 4500 J and 3000 J, ensuring the total remains constant.

An internal explosion is a sudden release of energy within an object that causes it to break apart with a force. In our exercise, such an explosion occurs, and the object splits into two pieces.
Understanding the mechanics of internal explosions helps explain why and how the pieces move. The explosion inside releases energy, often in the form of heat and work, causing the object’s segments to gain speed and fly apart.

• Momentum before the explosion is zero because the object starts from rest.
• Post-explosion, the conservation of momentum ensures that the momentum of one piece is balanced by the momentum of the other, but they travel in opposite directions.
• We calculated using the momentum principle \( m_1v_1 = m_2v_2 \), which shows how the velocities are connected to the masses.

This helps us distribute the 7500 J of energy across the two pieces using their mass ratio, as the explosion does not add or remove total momentum, only transforms the initial potential into kinetic action.