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Chapter 9: Problem 24

An object initially at rest breaks into two pieces as the result of an explosion. One piece has twice the kinetic energy of the other piece. What is the ratio of the masses of the two pieces? Which piece has the larger mass?

Short Answer

Expert verified
The mass ratio is 2:1, with the larger mass being the piece with less kinetic energy.

Step by step solution

01

Understand the Problem

We have an object initially at rest that explodes into two pieces. The pieces have different kinetic energies, where one piece has twice the kinetic energy of the other. We need to find the mass ratio of these pieces and which one is bigger.
02

Set Up the Kinetic Energy Equations

Let the masses of the two pieces be \( m_1 \) and \( m_2 \), and their velocities after the explosion be \( v_1 \) and \( v_2 \), respectively. Given that one piece has twice the kinetic energy of the other, we have \( KE_1 = \frac{1}{2}m_1v_1^2 \) and \( KE_2 = \frac{1}{2}m_2v_2^2 \). Suppose that \( KE_2 = 2 imes KE_1 \), leading to \( \frac{1}{2}m_2v_2^2 = 2 \times \frac{1}{2}m_1v_1^2 \).
03

Apply Conservation of Momentum

Since the object was initially at rest, its total momentum was zero. After the explosion, momentum conservation implies \( m_1v_1 = m_2v_2 \). This relationship allows us to express one velocity in terms of the other: \( v_2 = \frac{m_1}{m_2}v_1 \).
04

Solve the Kinetic Energy Equation

Substitute \( v_2 = \frac{m_1}{m_2}v_1 \) into the kinetic energy equation \( m_2v_2^2 = 2m_1v_1^2 \), giving us \( m_2 \left( \frac{m_1}{m_2}v_1 \right)^2 = 2m_1v_1^2 \). Simplify to get \( \left( \frac{m_1^2v_1^2}{m_2} \right) = 2m_1v_1^2 \), and further to \( m_1^2 = 2m_1m_2 \).
05

Rearrange and Solve for Mass Ratio

Divide both sides of \( m_1^2 = 2m_1m_2 \) by \( m_1 \), obtaining \( m_1 = 2m_2 \). Thus, the mass ratio \( \frac{m_1}{m_2} = 2 \).
06

Identify Which Piece Has the Larger Mass

From the mass ratio \( \frac{m_1}{m_2} = 2 \), \( m_1 = 2m_2 \), meaning piece 1 is heavier. Therefore, the piece with lesser kinetic energy, \( KE_1 \), is heavier because it has twice the mass.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Energy
Kinetic energy is the energy an object possesses due to its motion. For any object in motion, the kinetic energy can be calculated using the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the object.
In our explosion scenario, the total kinetic energy before the explosion is zero, as the object starts at rest.
After the explosion, the kinetic energy is distributed between the two pieces. The problem states that one piece has twice the kinetic energy of the other.
  • This can be expressed as \( KE_2 = 2 \times KE_1 \).
  • Using the kinetic energy formula, we have \( \frac{1}{2}m_2v_2^2 = 2 \times \frac{1}{2}m_1v_1^2 \).
This relationship not only helps us compare the kinetic energies but also plays a crucial role in solving for the mass ratio of the two pieces.
Mass Ratio
The mass ratio is a comparison of the masses of two objects. In the context of this exercise, understanding the mass ratio is essential to determining which piece of the exploded object is heavier.
Using the kinetic energy equations and conservation of momentum, we have two important expressions:
  • From conservation of momentum: \( m_1v_1 = m_2v_2 \), which implies \( v_2 = \frac{m_1}{m_2}v_1 \).
  • Substituting \( v_2 = \frac{m_1}{m_2}v_1 \) into the kinetic energy equation gives us the mass squared relationship: \( m_1^2 = 2m_1m_2 \).
After simplifying, we find the mass ratio \( \frac{m_1}{m_2} = 2 \). This solution indicates that the first mass \( m_1 \) is twice as heavy as the second mass \( m_2 \), which means \( m_1 = 2m_2 \).
Explosion Dynamics
Explosion dynamics refers to the study of the forces and energy transformations that occur during an explosion. In this context, the explosion causes the original object to break into two pieces with different kinetic energies.
To solve the problem, we utilize two key principles:
  • Conservation of Momentum: The total momentum before and after the explosion must remain the same. Since the object was at rest initially, the total momentum is zero and should remain zero post-explosion, leading to the equation \( m_1v_1 = m_2v_2 \).
  • Kinetic Energy Transformation: The source problem indicates that one part has twice the kinetic energy of the other. Together with momentum conservation, this allows us to deduce the velocity relationships and eventually the mass ratio.
By focusing on these dynamics, we unveil not only how energy is distributed between the fragments but also how the pieces' masses are related to each other. Solving these equations demonstrates the practical application of these fundamental physics concepts.

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Most popular questions from this chapter

A \(63-\mathrm{kg}\) canoeist stands in the middle of her \(22-\mathrm{kg}\) canoe. The canoe is \(3.0 \mathrm{m}\) long, and the end that is closest to land is \(2.5 \mathrm{m}\) from the shore. The canoeist now walks toward the shore until she comes to the end of the canoe. (a) When the canoeist stops at the end of her canoe, is her distance from the shore equal to, greater than, or less than \(2.5 \mathrm{m}\) ? Explain. (b) Verify your answer to part (a) by calculating the distance from the canoeist to shore.

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