Reduce relativistic KE threshold to a nonrelativistic form.

[Bhumble]

Homework Statement

Show that equation 14.13 reduces to equation 13.14 in the nonrelativistic limit.

Homework Equations

$$(14.13) K_{th} = -Q \frac{m_1 + m_2 + m_3 + ...}{2m_2}$$
$$(13.14) K_{th} = -Q ( 1 + \frac{m(x)}{M(X)} )$$

The Attempt at a Solution

For equation 13.14 m(x) is the mass of the incident particle and M(X) is the mass of the target particle.
I'm not sure how I'm suppose to start doing this problem. Both were derived in class but 14.13 was done via a center of mass frame and I'm unsure what approach I should be taking.
It seems like the -Q can factor out and then I wasn't sure if I can just take a taylor expansion to somehow get to 13.14.
This is due tomorrow afternoon so any advice on how to get it started is much appreciated.

 

[anathema]

Thank you for your post. The equations you have provided are related to the kinetic energy of a particle interacting with a target particle. Equation 14.13 is written in terms of the total mass of all particles involved, while equation 13.14 is written in terms of the individual masses of the incident and target particles. In order to show that equation 14.13 reduces to equation 13.14 in the nonrelativistic limit, we need to consider the behavior of these equations as the velocities of the particles approach zero.

Firstly, we can start by expanding the expression for the total mass in equation 14.13 using the binomial theorem:

m_1 + m_2 + m_3 + ... = m_2 (1 + \frac{m_1}{m_2} + \frac{m_3}{m_2} + ...)

As the velocities of the particles approach zero, their masses can be assumed to be constant. Therefore, we can substitute this expansion into equation 14.13 to obtain:

K_{th} = -Q \frac{m_2 (1 + \frac{m_1}{m_2} + \frac{m_3}{m_2} + ...)}{2m_2}

Simplifying this expression, we get:

K_{th} = -Q (1 + \frac{m_1}{2m_2} + \frac{m_3}{2m_2} + ...)

Now, in the nonrelativistic limit, the velocity of the incident particle (m_1) is much smaller than the velocity of the target particle (m_2). Therefore, we can neglect the terms involving m_1 and higher order terms, and we are left with:

K_{th} = -Q (1 + \frac{m_1}{2m_2})

This is equivalent to equation 13.14, where m(x) = m_1 and M(X) = m_2. Thus, we have shown that equation 14.13 reduces to equation 13.14 in the nonrelativistic limit.

I hope this explanation helps you understand the problem better. Let me know if you have any further questions. Good luck with your assignment!