Lorentz momentum: A proton of mass m is accelerated up

[squelch]

Homework Statement

A proton of mass "m" is accelerated up to a kinetic energy "K" and then collides with a stationary proton at rest. All that is left after the collision is a new particle of mass "M".
(a) Write out the momentum and energy equations for the collision.
(b) What is the maximum mass M that can be created in this collision?

Homework Equations

I'm not quite sure what to use here. I have:

$$\begin{array}{l}<br /> {E^2} = {p^2}{c^2} + {m^2}{c^4}\\<br /> p = \gamma mv<br /> \end{array}$$

The Attempt at a Solution

It seems like what I'm supposed to do is use the energy-momentum relationship and the formula for kinetic energy in this way:

$$\begin{array}{l}<br /> {(K - m{c^2})^2} = {p^2}{c^2} + {m^2}{c^4}\\<br /> K - m{c^2} = \sqrt {{p^2}{c^2} + {m^2}{c^4}}<br /> \end{array}$$

From here, do I use $K = \frac{1}{2}m{v^2}$ to put together the $p=mv$ relationship?

edit: I'm supposing that what I'm supposed to do here is solve for $v$ somehow.

 

[vela]

Homework Statement

A proton of mass "m" is accelerated up to a kinetic energy "K" and then collides with a stationary proton at rest. All that is left after the collision is a new particle of mass "M".
(a) Write out the momentum and energy equations for the collision.
(b) What is the maximum mass M that can be created in this collision?

Homework Equations

I'm not quite sure what to use here. I have:

$$\begin{array}{l}<br /> {E^2} = {p^2}{c^2} + {m^2}{c^4}\\<br /> p = \gamma mv<br /> \end{array}$$

The Attempt at a Solution

It seems like what I'm supposed to do is use the energy-momentum relationship and the formula for kinetic energy in this way:

$$\begin{array}{l}<br /> {(K - m{c^2})^2} = {p^2}{c^2} + {m^2}{c^4}\\<br /> K - m{c^2} = \sqrt {{p^2}{c^2} + {m^2}{c^4}}<br /> \end{array}$$

From here, do I use $K = \frac{1}{2}m{v^2}$ to put together the $p=mv$ relationship?

edit: I'm supposing that what I'm supposed to do here is solve for $v$ somehow.

You can't use the Newtonian expressions for the kinetic energy and momentum. You have to use the relativistic expressions.

A word of advice: Write your equations in terms of energies, momenta, and the masses. Your variables should be E's, p's, and m's, like in the first relevant equation you wrote above.

Before the collision, you have the energy and momentum of the first proton and the energy and momentum of the second proton. After the collision, you have the energy and momentum of the new particle. Assign a variable to each of those quantities and then write down the equations for conservation of momentum and energy.