Special Relativity -- Elastic Particle Collision Algebra

In summary, the conversation discusses a head-on elastic collision between two particles with different rest masses. The laws of conservation of total energy and momentum are applied to find the resulting speeds of the particles after the collision. The equations for conservation of mass and momentum are used, but there are two unknowns in the resulting equations. However, it is possible to express one unknown in terms of the other, allowing for the determination of both resulting speeds.
  • #1
aamirza
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Homework Statement



Consider the following head-on elastic collision. Particle 1 has rest mass 2mo, and particle 2 has rest mass mo. Before the collision, particle 1 movies toward particle 2, which is initially at rest, with speed u (= 0.600c ). After the collision each particle moves in the forward direction with speeds of u1 and u2, respectively.

a) Apply the laws of conservation of total energy (or, equivalently, of relativistic mass) and of relativistic momentum to this collision and solve the resulting equation to find u1 and u2 (the resulting speeds of the two particles).

Homework Equations



Well, I know the mass is conserved in elastic collisions.

##m_1\gamma_1 + m_2\gamma_2 = m_1\gamma_{1}^{'} + m_2\gamma_{2}^{'} ##

where gamma is the Lorrentz factor. I also know energy is conserved, (Ei = Ef, where E = Moc2 + (M - Mo)c2 where Mois the rest mass and M is the relativistic mass), but that basically reduces to the same thing as the mass equation.

I also know the equation for the conservation of momentum,

##p_1 + p_2 = p_{1}^{'} + p_{2}^{'} ##

where

##p = M_o\gamma u ##

Mo is the rest mass and u is the speed.

The Attempt at a Solution



So first I plugged in the values of the masses and speeds into the mass-conservation equation and got

##2m_o\gamma_1 + m_o\gamma_2 = 2m_o\gamma_1^{'} + m_o\gamma_2^{'} ##

##3.5 - 2\gamma_1^{'} = \gamma_2^{'} ##

but when I got around to plugging the numbers into the momentum equation, I got

## 2m_o\gamma_1u_1 = 2m_o\gamma_{1}^{'}u_{1}^{'} + m_o\gamma_{2}^{'}u_{2}^{'} ##

## 2\gamma_1u_1 = 2\gamma_{1}^{'}u_{1}^{'} + \gamma_{2}^{'}u_{2}^{'} ##

The problem arises when, no matter what I substitute in for any of the Lorrentz factors (gammas), I always get two unkonws in the equation, u1 prime and u2 prime. I can't think of a way to isolate just for one. Any help?
 
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  • #2
The concept of relativistic mass is not used any more in physics.

You have two equations, it is fine to have two unknowns.
##\gamma_1'## and ##u_1'## have a known relation, you can use one of them to express the other (same for 2 of course).
 
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Related to Special Relativity -- Elastic Particle Collision Algebra

1. What is special relativity?

Special relativity is a theory in physics that describes how objects move in relation to each other at high speeds, near the speed of light. It is based on the idea that the laws of physics are the same for all observers moving at a constant velocity.

2. How does special relativity apply to elastic particle collisions?

In elastic particle collisions, the total energy of the system is conserved. This means that the total kinetic energy and momentum of the particles before and after the collision remain the same. Special relativity takes into account the effects of high speeds on these quantities and allows for accurate calculations of elastic collisions involving particles traveling at relativistic speeds.

3. What is the algebra involved in special relativity elastic particle collisions?

The algebra involved in special relativity elastic particle collisions is based on the equations of special relativity, including the Lorentz transformations, which describe how time, length, and mass are affected by high speeds. By using these equations, we can calculate the velocities, energies, and momenta of particles before and after a collision.

4. Are there any limitations to special relativity elastic particle collision algebra?

Special relativity elastic particle collision algebra is limited to systems that involve particles moving at speeds close to the speed of light. It also assumes that the particles involved are point-like and have no internal structure. In more complex systems, such as those involving particles with internal structure, other theories, such as quantum mechanics, must be used.

5. Why is special relativity important in understanding elastic particle collisions?

Special relativity is important in understanding elastic particle collisions because it provides a framework for accurately describing the behavior of particles at high speeds. Without taking into account the effects of special relativity, the calculations for elastic collisions at relativistic speeds would be inaccurate. Special relativity also helps us understand the relationship between energy, mass, and momentum in these collisions.

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