Choosing the Correct Momentum Equation: Newtonian vs Relativistic

[Einstein's Cat]

I have come to notice that there are two equations for calculating momentum and I am under the impression that both equations provide different answers.

There is the Newtonian, classical equation of p = mv, where p, is momentum, m is mass, and v is velocity.

Yet also there is the relativistic equation for calculating momentum of p= mv / √ 1 - v squared / c squared, where c is the speed of light.

Therefore, what equation would you recommend to use, and which equation is more accurate?

Thank you for your help and happy holidays!

 

[Nosebgr]

It depends. If you are working with particles traveling at relativistic speeds, then you would need to use the relativistic equation. For particles traveling at velocities much smaller than the speed of light, then you can use the Newtonian approximation. If you want, you can compare the results of two equations for slow moving particles and see for yourself that the results are pretty much exactly the same.

 

[Orodruin]

The Newtonian p = mv is an approximation of the relativistic result and is accurate for velocities much smaller than the speed of light. If you are not dealing with objects traveling at relativistic velocities, you will do just fine applying the Newtonian version.

p= mv / √ 1 - v squared / c squared

Just as a heads-up, writing "squared" in an equation tends to severely limit the readability. If you are not yet familiar with writing equations in LaTeX, I suggest you simply use ^ when referring to an exponent. You could also do with a few parentheses. This expression would be much more readable if you wrote it as "p = mv/√(1- v^2/c^2)".

 

[Nugatory]

@Einstein's Cat It would be a good exercise to try calculating the momentum both ways for a few different objects:
- a thrown stone: v=30 meters/sec
- a cannonball: v=300 meters/sec
- a spaceship in orbit: v=9000 meters/sec
- mass-extinction meteorite: v=30000 meters/second

Do this and you'll understand why we still use and teach Newtonian physics.