particle accelerator in space

Question

I'm attempting to learn special relativity and i'm having trouble calculating velocity and momentum for each part of the system after interactions.

I wanted to know how fast a linear accelerator and particle would move with respect to the rest frame after a period of acceleration assuming the system is placed in free space. I had some trouble understanding how to apply acceleration to a particle at rest that would result in relativistic speeds so I took the energy approach. If we imagine the accelerator can accelerate a proton to 1TeV of kinetic energy we get the velocity for the proton given:

$$ KE = \frac{m_0c^2}{\sqrt{1-\frac{v^2}{c^2}}} $$ solving for v: $$ v = \sqrt{1- (\frac{m_0c^2}{KE})^2}c $$

and the velocity for the accelerator given:

$$ KE = \frac{1}{2}mv^2 $$ solving for v: $$ v = \sqrt{\frac{2KE}{m}} $$

When I plugged in the numbers and double checked the energy I found that: $$ \frac{1}{2}m_av_a^2 = (\gamma(v_p) -1)m_0c^2 $$ where the left hand represents the kinetic energy of the accelerator and the right hand represents the kinetic energy of the proton. So at this point I felt pretty confident in my calculations for the velocities, but when I looked at the momentum I discovered that something was terribly wrong. The momentum of the accelerator $m_av_a$ definitely was not equal to the momentum of the proton $\gamma m_0v$.

Ultimately I wanted to look at inelastic collisions between relativistic and non-relativistic systems, but I wanted to be able to to do the calculations from rest to rest(which should be easy in a closed system) and i seem to be stumped. So what am I doing wrong? Do I need to be approaching this from another frame? How should I calculate momentum and velocity correctly.

Answer 1

It seems that problem is you're mixing non-relativistic and relativistic expressions. For general collision problems in special relativity, you should use the relativistic expressions for energy $E$ and momentum $p$ for both objects; $$ E = \gamma m c^2, \qquad p = \gamma m v $$ and then use conservation of relativistic energy and/or momentum depending on the context. In this case, the kinetic energy of a certain object is given. Relativistically, kinetic energy is defined as total energy minus rest energy; $$ K = \gamma mc^2 - mc^2 = (\gamma-1)mc^2 $$ You can use this expression for the kinetic energy of the accelerated particle to obtain the speed of the accelerated particle. Then, you can use conservation of relativistic momentum to determine the speed of the accelerator. Let me know if you'd like more detail.

Cheers!