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Chapter 9: Problem 35

Two objects moving with a speed \(v\) travel in opposite directions in a straight line. The objects stick together when they collide, and move with a speed of \(v / 4\) after the collision. (a) What is the ratio of the final kinetic energy of the system to the initial kinetic energy? (b) What is the ratio of the mass of the more massive object to the mass of the less massive object?

Short Answer

Expert verified
(a) The kinetic energy ratio is \( \frac{1}{16} \). (b) The mass ratio is \( \frac{5}{3} \).

Step by step solution

01

Define Initial Conditions

Let the masses of the two objects be \( m_1 \) and \( m_2 \), and we know both are moving with speed \( v \) in opposite directions. The initial velocity of object 1 is \( v \) and for object 2 is \(-v\).
02

Calculate Initial Kinetic Energy

The initial kinetic energy of the system is the sum of the kinetic energies of both objects. This can be calculated using:\[ KE_{initial} = \frac{1}{2} m_1 v^2 + \frac{1}{2} m_2 v^2 = \frac{1}{2} (m_1 + m_2) v^2 \]
03

Calculate Final Kinetic Energy

After collision, the objects stick together and move with speed \( \frac{v}{4} \). The final mass is \( m_1 + m_2 \). The final kinetic energy is:\[ KE_{final} = \frac{1}{2} (m_1 + m_2) \left(\frac{v}{4}\right)^2 = \frac{1}{2} (m_1 + m_2) \frac{v^2}{16} \]
04

Evaluate Ratio of Kinetic Energies

The ratio of the final kinetic energy to initial kinetic energy is:\[ \frac{KE_{final}}{KE_{initial}} = \frac{\frac{1}{2} (m_1 + m_2) \frac{v^2}{16}}{\frac{1}{2} (m_1 + m_2) v^2} = \frac{1}{16} \]
05

Use Conservation of Momentum

Conservation of momentum gives us:\[ m_1 v - m_2 v = (m_1 + m_2) \frac{v}{4} \] or \[v(m_1 - m_2) = \frac{v}{4}(m_1 + m_2)\].
06

Simplification and Solve for Mass Ratio

Canceling \(v\) and rearranging gives:\[ 4(m_1 - m_2) = (m_1 + m_2) \]\[ 4m_1 - 4m_2 = m_1 + m_2 \]\[ 3m_1 = 5m_2 \]\[ \frac{m_1}{m_2} = \frac{5}{3} \]
07

Conclusion on Mass Ratio

The more massive object is \( \frac{5}{3} \) times the mass of the less massive object, satisfying the ratio found from the momentum conservation equation.

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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 that an object possesses due to its motion. This plays a crucial role in understanding the behavior of objects during a collision. In our scenario, two objects are moving towards each other with speed \( v \), each carrying kinetic energy. Kinetic energy in physics is calculated using the formula:
\[ KE = \frac{1}{2} mv^2 \]
Here, \( m \) represents mass and \( v \) is the velocity of the object.
  • Initially, both objects have kinetic energy, and we sum these to find the total initial kinetic energy of the system.
  • After the collision, the objects stick together, and their speed reduces to \( \frac{v}{4} \).
  • This decrease in speed results in less kinetic energy post-collision.
    Therefore, the ratio of final kinetic energy to initial kinetic energy can be calculated as shown:
    \[ \frac{KE_{final}}{KE_{initial}} = \frac{1}{16} \]
    This shows a significant reduction in kinetic energy after the objects collide and move more slowly as one unit.
    • Momentum Conservation
    Momentum is a key concept when analyzing collisions, where it reflects the tendency of an object to continue moving in its direction. Momentum for a single object is computed as:
    \[ p = mv \]
    In a system of moving objects, the total momentum is the sum of the momentum of each object. Importantly, momentum is directional, meaning if an object moves in the opposite direction, its momentum is subtracted.
  • In our exercise, initially, object 1 moves right with momentum \( m_1v \), while object 2 moves left with momentum \(-m_2v \).
  • The law of conservation of momentum states the total momentum before the collision equals the total momentum after.
    The conservation equation is:
    \[ m_1v - m_2v = (m_1 + m_2) \frac{v}{4} \]
    By solving this equation, we can deduce other qualities like the relation between the masses, described further below.
    • Mass Ratio
    In the context of a collision, analyzing the masses of colliding objects provides valuable insights. Mass ratio refers to the relation between the masses of the two colliding objects, revealing the dynamic distribution of the system.
  • Initially, using momentum conservation, we established the equation linking the masses:
    \[ 4(m_1 - m_2) = (m_1 + m_2) \]
    Solving this leads to finding the mass ratio:
    \[ 3m_1 = 5m_2 \]
  • The solution tells us \( \frac{m_1}{m_2} = \frac{5}{3} \).
    This ratio means that the more massive object is \( \frac{5}{3} \) times the mass of the less massive object. Such understanding clarifies how mass influences movement and speed in post-collision scenarios.

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    Most popular questions from this chapter

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