Problem 100 - University Physics with Modern Physics

Question

Energy Sharing. An object with mass \(m,\) initially at rest, explodes into two fragments, one with mass \(m_{A}\) and the other with mass \(m_{B},\) where \(m_{A}+m_{B}=m .\) (a) If energy \(Q\) is released in the explosion, how much kinetic energy does each fragment have immediately after the explosion? (b) What percentage of the total energy released does each fragment get when one fragment has four times the mass of the other?

Step-by-Step Solution

Answer

Fragment with larger mass gets 80%, and the smaller fragment gets 20% of the energy.

Step 1: Understand Conservation of Momentum

Initially, the object is at rest, so total momentum is 0. After the explosion, the momentum must still be 0. If fragment A has velocity \(v_A\) and fragment B has velocity \(v_B\), then the equation is:\[m_A v_A = -m_B v_B\]

Step 2: Relate Velocities to Kinetic Energy

Using the conservation of momentum, we can find the ratio of velocities:\[\frac{v_A}{v_B} = -\frac{m_B}{m_A}\]The kinetic energy for each fragment is given by:\[K_A = \frac{1}{2} m_A v_A^2\quad \text{and} \quad K_B = \frac{1}{2} m_B v_B^2\]

Step 3: Use Energy Release Relation

Since the total energy released is \(Q\), the sum of the kinetic energies must equal \(Q\):\[K_A + K_B = Q\]

Step 4: Solve for Kinetic Energies

We know from conservation of momentum that \(v_A = \frac{m_B}{m_A} v_B\). Substitute this into the expression for \(K_A\):\[K_A = \frac{1}{2} m_A \left(\frac{m_B}{m_A} v_B\right)^2 = \frac{1}{2} \frac{m_B^2}{m_A} v_B^2\]Substituting \(K_B = \frac{1}{2} m_B v_B^2\) into the energy relation, we get:\[\frac{1}{2} \frac{m_B^2}{m_A} v_B^2 + \frac{1}{2} m_B v_B^2 = Q\]Solve to find:\[ v_B^2 = \frac{2Q}{m_B + \frac{m_B^2}{m_A}} \]Substitute back into equations for \(K_A\) and \(K_B\):\[K_A = \frac{Q m_B}{m} \quad \text{and} \quad K_B = \frac{Q m_A}{m}\]

Step 5: Calculate Energy Distribution with Mass Ratio

When one fragment is four times the mass of the other, let \(m_A = 4m_B\). Then, \(m_A + m_B = m = 5m_B\). Substitute into the expressions for kinetic energies: \[K_A = Q \frac{m_B}{5m_B} = \frac{Q}{5} \quad \text{and} \quad K_B = Q \frac{4m_B}{5m_B} = \frac{4Q}{5}\]

Step 6: Determine Percentage of Total Energy

The percentage of total energy for each fragment can be calculated as:\[\text{Percent of } K_A = \left( \frac{Q/5}{Q} \right) \times 100 = 20\%\]\[\text{Percent of } K_B = \left( \frac{4Q/5}{Q} \right) \times 100 = 80\%\]

Key Concepts

Kinetic Energy

Kinetic energy is a measure of the energy an object possesses due to its motion. When dealing with explosions or collisions, it's crucial to understand how this energy is distributed among the resulting fragments. For any object with mass, its kinetic energy (K) is calculated using the formula:\[K = \frac{1}{2} mv^2\]where \(m\) is the mass and \(v\) is the velocity of the object. When an object explodes, each fragment moves with a certain velocity, and thus has its own kinetic energy. In the context of the original exercise, when the object explodes, the total kinetic energy of the fragments sums up to the energy released, \(Q\). This conservation ensures that none of the energy is lost in the process but instead redistributed into kinetic forms for the fragments. Understanding and calculating this distribution helps to determine how much energy each fragment gains post-explosion.

Energy Distribution

In any event involving energy release, such as an explosion, it's vital to know how the energy is shared among the products of the event. This sharing is referred to as energy distribution. Different masses will move with different velocities and, consequently, have different kinetic energies. As per the original exercise, if one fragment has a mass four times larger than the other, energy distribution follows specific rules:

• The larger mass moves slower, acquiring less kinetic energy in proportion to its mass.
• The smaller mass moves faster, taking a larger share of the kinetic energy.

This results in the smaller fragment receiving more energy, despite its smaller size. Expressing it percentage-wise can clarify the distribution of total energy, as seen when one fragment with four times the mass uses 20% of energy and the other 80%. This asymmetric distribution highlights how mass differences affect energy outcomes post-explosion.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed—only transformed from one form to another. This fundamental law governs all physical processes, including explosions and collisions. In the original problem's context, the total energy released during the explosion, represented as \(Q\), is completely transformed into the kinetic energy of the resulting fragments. The sum of individual kinetic energies after the explosion must equal the energy \(Q\) originally discharged.This principle ensures that regardless of how many fragments result, or their speeds, if you add up all their kinetic energies, you'll always account for the entire initial energy. Hence, the conservation of energy offers a reliable way to analyze and calculate outcomes in dynamics problems by maintaining equilibrium in energy transformation.