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Mobile App Chapter 35: Problem 54
Consider a one-dimensional collision at relativistic speeds between two particles with masses \(m_{1}\) and \(m_{2}\). Particle 1 is initially moving with a speed of \(0.700 c\) and collides with particle \(2,\) which is initially at rest. After the collision, particle 1 recoils with speed \(0.500 c\), while particle 2 starts moving with a speed of \(0.200 c\). What is the ratio \(m_{2} / m_{1} ?\)
Short Answer
Expert verified
Answer: The approximate mass ratio of the two relativistic particles after the collision is 1.964.
Step by step solution
01
Write down the conservation of momentum equation for relativistic speeds
In general, the conservation of momentum states that the total initial momentum is equal to the final momentum. For particles moving at relativistic speeds, we need to take into account the relativistic mass and velocity. The relativistic momentum can be expressed as:
\(P_{rel} = \frac{m_0v}{\sqrt{1-\frac{v^2}{c^2}}}\)
where \(m_0\) is the rest mass, \(v\) is the velocity, and \(c\) is the speed of light.
For this problem, since particle 2 is initially at rest, the initial momentum is only due to particle 1:
\(P_{initial} = P_{1, initial} + P_{2, initial} = \frac{m_1 v_1}{\sqrt{1-\frac{v_1^2}{c^2}}}\)
and the final momentum is due to both particles:
\(P_{final} = P_{1, final} + P_{2, final} = \frac{m_1 v_{1, final}}{\sqrt{1-\frac{v_{1, final}^2}{c^2}}} + \frac{m_2 v_{2, final}}{\sqrt{1-\frac{v_{2, final}^2}{c^2}}}\)
Since the momentum is conserved, we have:
\(P_{initial} = P_{final}\)
02
Substitute the given values and solve for the mass ratio
We are given \(v_1 = 0.700c\), \(v_{1, final} = 0.500c\), and \(v_{2, final} = 0.200c\). Now we can substitute these values into the conservation of momentum equation and get:
\(\frac{m_1 (0.700c)}{\sqrt{1-\frac{(0.700c)^2}{c^2}}} = \frac{m_1 (0.500c)}{\sqrt{1-\frac{(0.500c)^2}{c^2}}} + \frac{m_2 (0.200c)}{\sqrt{1-\frac{(0.200c)^2}{c^2}}}\)
Now, we can simplify the equation by cancelling the speed of light \(c\) in the numerators and denominators:
\(\frac{m_1 (0.700)}{\sqrt{1-0.490}} = \frac{m_1 (0.500)}{\sqrt{1-0.250}} + \frac{m_2 (0.200)}{\sqrt{1-0.040}}\)
We can further simplify the equation by calculating the values inside the square roots:
\(\frac{m_1 (0.700)}{\sqrt{0.510}} = \frac{m_1 (0.500)}{\sqrt{0.750}} + \frac{m_2 (0.200)}{\sqrt{0.960}}\)
Now, we can divide by \(m_1\) on both sides of the equation:
\(\frac{0.700}{\sqrt{0.510}} = \frac{0.500}{\sqrt{0.750}} + \frac{m_2 (0.200)}{m_1 \sqrt{0.960}}\)
Let's denote the mass ratio as \(R = \frac{m_2}{m_1}\). Then we can write:
\(R = \frac{m_2}{m_1} = \frac{\frac{0.700}{\sqrt{0.510}} - \frac{0.500}{\sqrt{0.750}}}{\frac{0.200}{\sqrt{0.960}}}\)
03
Calculate the mass ratio
Lastly, we need to calculate the mass ratio \(R\):
\(R = \frac{\frac{0.700}{\sqrt{0.510}} - \frac{0.500}{\sqrt{0.750}}}{\frac{0.200}{\sqrt{0.960}}} \approx 1.964\)
So the mass ratio \(m_2/m_1 \approx 1.964\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
One-Dimensional Collision
Understanding a one-dimensional collision involves analyzing the interaction between two objects moving along a single straight line. In our case, this interaction is a collision between two particles at relativistic speeds. When we observe particles moving at significant fractions of the speed of light, we must adapt our approach to factor in the effects of special relativity. Here, the primary concern is to determine how the momentum is transferred between the particles during the collision, and how this affects their subsequent velocities.
It's also essential to note that in one-dimensional collisions, particles can only move forwards or backwards along the line of their motion, which simplifies the analysis. The complexity of involving relativistic effects transforms a classically straightforward calculation into a more nuanced problem, necessitating the use of relativistic formulas for both momentum and mass.
It's also essential to note that in one-dimensional collisions, particles can only move forwards or backwards along the line of their motion, which simplifies the analysis. The complexity of involving relativistic effects transforms a classically straightforward calculation into a more nuanced problem, necessitating the use of relativistic formulas for both momentum and mass.
Relativistic Mass
In the realm of high speeds approaching the speed of light, mass can no longer be considered constant. The concept of relativistic mass emerges from the observation that an object's inertia increases as its velocity approaches the speed of light, denoted by the symbol 'c'. This means that as an object moves faster, it behaves as if its mass is increasing.
The equation for relativistic mass is given by \( m_{rel} = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}} \), where \( m_0 \) is the rest mass of the object, \( v \) is the velocity of the object, and \( c \) is the speed of light. This equation shows that relativistic mass depends not only on the object’s intrinsic rest mass but also on its velocity relative to the observer.
The equation for relativistic mass is given by \( m_{rel} = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}} \), where \( m_0 \) is the rest mass of the object, \( v \) is the velocity of the object, and \( c \) is the speed of light. This equation shows that relativistic mass depends not only on the object’s intrinsic rest mass but also on its velocity relative to the observer.
Conservation of Momentum
Conservation of momentum is a fundamental principle in physics that holds true in both classical and relativistic contexts. It asserts that the total momentum of a closed system remains constant if no external forces act upon it. In a closed system, the sum of the momentum of all objects before a collision equals the sum after the collision.
This pivotal law allows us to predict the outcome of collisions, such as the one in our problem. For relativistic speeds, we employ the relativistic momentum formula, considering the increased inertia encapsulated by the concept of relativistic mass. The formula for relativistic momentum, as applied in our scenario, ensures that the total momentum is conserved across the initial and final states of the collision.
This pivotal law allows us to predict the outcome of collisions, such as the one in our problem. For relativistic speeds, we employ the relativistic momentum formula, considering the increased inertia encapsulated by the concept of relativistic mass. The formula for relativistic momentum, as applied in our scenario, ensures that the total momentum is conserved across the initial and final states of the collision.
Speed of Light
The speed of light in a vacuum, symbolized as \( c \), is a universal physical constant crucial to the theory of relativity and many areas of physics. It is approximately \( 3 \times 10^8 \) meters per second. When we discuss relativistic speeds, we're referring to speeds that are a significant fraction of the speed of light.
At such extreme velocities, classical mechanics gives way to relativistic physics. The effects, like time dilation and length contraction, become noticeable and must be considered in calculations. In our exercise, velocities are given as a fraction of \( c \) to indicate their relativistic nature, influencing how momentum and mass are perceived and calculated.
At such extreme velocities, classical mechanics gives way to relativistic physics. The effects, like time dilation and length contraction, become noticeable and must be considered in calculations. In our exercise, velocities are given as a fraction of \( c \) to indicate their relativistic nature, influencing how momentum and mass are perceived and calculated.
Mass Ratio
The mass ratio in the context of our collision problem is defined as the ratio of the two particles' masses: \( m_2/m_1 \) after the collision occurs. It's a measure of how the mass is distributed between the two particles after their relativistic encounter. Through the conservation of momentum, we can relate this ratio to the velocities of the particles both before and after the collision.
Calculating this ratio requires careful manipulation of the relativistic momentum equations for both particles. When dealing with relativistic scenarios, the mass ratio can reveal insights about the rest masses of the particles involved, especially in high-energy physics experiments where particles are often accelerated to near-light speeds.
Calculating this ratio requires careful manipulation of the relativistic momentum equations for both particles. When dealing with relativistic scenarios, the mass ratio can reveal insights about the rest masses of the particles involved, especially in high-energy physics experiments where particles are often accelerated to near-light speeds.
