Problem 90 - University Physics with Modern Physics

Question

A 7.0 -kg shell at rest explodes into two fragments, one with a mass of 2.0 \(\mathrm{kg}\) and the other with a mass of 5.0 \(\mathrm{kg}\) . If the heavier fragment gains 100 \(\mathrm{J}\) of kinetic energy from the explosion, how much kinetic energy does the lighter one gain?

Step-by-Step Solution

Answer

The lighter fragment gains 100 J of kinetic energy.

Step 1: Determine Total Energy Conservation

In an explosion, the total kinetic energy gained by the fragments is equal to the sum of the kinetic energies of each fragment. Since energy is conserved, the kinetic energy gained by the 5.0 kg fragment plus the kinetic energy gained by the 2.0 kg fragment will equal the total initial kinetic energy gained from the explosion.

Step 2: Identify Known Values

We know the kinetic energy gained by the 5.0 kg fragment is 100 J. Since the shell is initially at rest, the entire system starts with a kinetic energy of 0 J.

Step 3: Calculate Kinetic Energy of Lighter Fragment

Since the total kinetic energy gained by the system must remain zero initially, but transforms completely into kinetic energy post-explosion, the kinetic energy gained by the 2.0 kg fragment must counterbalance the kinetic energy obtained by the heavier fragment. Therefore, it gains 100 J as well because the system's net kinetic energy must reflect conservation in distribution.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It's a fundamental concept in physics related to how moving objects interact. The formula to calculate kinetic energy is given by:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass of the object and \( v \) is its velocity. In the context of explosions, kinetic energy is an essential part of understanding how the energy released is distributed among fragments.
When the shell in the exercise explodes, its total mass breaks into two parts. Each fragment gains kinetic energy according to its mass and velocity post-explosion. Since the system initially has no kinetic energy (the shell is at rest), the energy released becomes kinetic energy for the fragments.

Explosions

Explosions involve rapid expansion and release of energy, often transforming potential energy stored within an object into kinetic energy. In physics, analyzing explosions helps us understand how energy distributes across moving pieces.

• An explosion is often unpredictable; however, the conservation of energy applies. This means that all the initial energy transforms into kinetic energy in some manner.
• In our exercise, the shell explosion results in two fragments, each moving rapidly. Importantly, there's no other energy (like sound or heat) considered in this simplified scenario.
• The key to solving this problem is understanding that the energy initially at zero transforms fully into the kinetic energy of fragments. Thus, the kinetic energy must be proportional to the mass and speed each fragment achieves after detonation.

Energy Transformation

Energy transformation refers to the process of converting energy from one form to another. During an explosion, potential energy transforms into kinetic energy—energy due to motion. The conservation of energy principle stresses that in a closed system, energy cannot be created or destroyed, only transformed.

• This principle explains why the total kinetic energy acquired by the fragments must equal the energy initially released. In simpler terms, one fragment's gain reflects the other's need for an entirely new distribution of energy post-explosion.
• In the exercise, the problem follows this principle by ensuring both fragments receive the total energy output of the explosion equally, because their combined energy equates to what the shell initially mustered from being at rest to its newfound kinetic state.
• The lighter fragment therefore gains as much kinetic energy as the heavier one, maintaining the equilibrium needed for conservation within the explosion framework.
• Recognizing these transformations helps us solve real-world issues related to dynamics and energetic exchanges both in controlled and natural explosion scenarios.