91Ó°ÊÓ

Kinetic energy is the energy that an object possesses due to its motion. It depends on two main factors: mass and velocity. For an object with mass \( m \) and velocity \( v \), the kinetic energy \( KE \) is given by the formula:\[ KE = \frac{1}{2} m v^2 \]In the scenario provided, the initial kinetic energy of the 20 kg body is calculated before the explosion using the given velocity of 200 m/s. This results in an initial kinetic energy of 400,000 J. Once the explosion occurs, the body breaks into three parts, each of which has its own kinetic energy calculated using the same formula based on their respective velocities and masses. The significant change in kinetic energy after the explosion indicates the energy released during this event.

Understanding vector components is crucial when dealing with motion in different directions. Each vector, like momentum or velocity, can be broken down into components along the perpendicular axes, usually denoted by \( x \) and \( y \) directions.
In our exercise, we have initial momentum in just the \( x \)-direction since the body is moving only in that direction initially. After the explosion, the momentum of the parts needs to be described using vector components. For example:

• The 10 kg part moves in the positive \( y \)-direction, so its momentum only has a \( y \)-component: (0, 1000) kg m/s.
• The 4 kg part moves in the negative \( x \)-direction, giving it a momentum vector of (-2000, 0) kg m/s.

Breaking vectors into components helps apply the principle of momentum conservation more straightforwardly in each direction.

Momentum conservation is a fundamental principle stating that in a closed system, the total momentum before an event (like an explosion or collision) must equal the total momentum after the event. This principle can be mathematically represented by:\[ \vec{p}_{initial} = \vec{p}_{final} \]For the exercise, initially, the momentum is only in the \( x \) direction, calculated to be 4000 kg m/s. After the body explodes, we must consider this initial momentum when accounting for the new momentum vectors of the three pieces. The total momentum is conserved in both the \( x \)- and \( y \)- directions. Solving for the third part’s momentum using this conservation principle helps find its velocity vector \((v_{3x}, v_{3y}) = (1000, -500) \) m/s. This calculation verifies that the total momentum after the explosion equals the initial momentum.

An internal explosion refers to an event where an object splits apart due to internal forces, like the explosive force breaking the body into multiple pieces. An essential aspect of this process is how the momentum and energy transformations occur.
During an internal explosion, the momentum conservation law ensures that the total system momentum remains unchanged despite the massive increase in kinetic energy. However, because the system energy is not conserved (due to energy conversion from stored potential energy into kinetic energy), the kinetic energy increases significantly, as evidenced in this problem.

• The original state features a single momentum vector and a total kinetic energy of 400,000 J.
• Following the explosion, the system’s total kinetic energy increases to 4,300,000 J. This massive increase (an additional 3,900,000 J) quantifies the energy released as kinetic energy in the explosion, illustrating how massive energy transformations occur during such events.