# Comparing relativistic momentum to classical

PsychonautQQ
EDIT: Okay I don't expect an answer for this because of my crappy attempt at LaTex, i'll work on making it look prettier sorry

## Homework Statement

If the kinetic energy of a particle is equal to twice its rest energy, what percentage error is made by using p = mu for the magnitude of its momentum?

## Homework Equations

$$$E_i=mc^2$$$
$$$p = \frac{mu}{(1-\frac{u^2}{c^2})^\frac{1}{2}}$$$
$$$p = mu$$$
$$$K=\frac{p^2}{2m}$$$

## The Attempt at a Solution

I set K = 2mc^2 and then solved for the relative velocity u and ended up with u=(2/sqrt(5))*c

I then set K = 2mc^2 again but this time the momentum term was non relativistic, and solving for u I got u = 2c

now i'm lost

Mentor
Try using your two equations for momentum to get the ratio of the relativistic momentum to the pre-relativistic momentum. If you did the algebra right so far, you should get the right answer.

Gold Member
2021 Award
Just perform a Taylor expansion in the quantity $p/(mc)$, which is small for non-relativistic motion. Here $m$ is the invariant (rest) mass of the particle. One should not use any other masses in relativistic physics anymore. That's outdated since 1908 when Minkowski figured out the mathematical structure of special-relativistic space-time!

Mentor
Another equation you might consider using is K=(γ-1)mc2. This would greatly simplify the analysis.

Chet

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