Is This Equation Correct? p² = p₀² + 2mΔk

  • Thread starter Thread starter Superstring
  • Start date Start date
AI Thread Summary
The equation p² = p₀² + 2mΔk is derived from kinetic energy and momentum relationships, confirming its correctness for speeds much lower than the speed of light. The derivation utilizes the relationships between kinetic energy (k = ½mv²) and momentum (p = mv), leading to a dimensionally valid equation. It simplifies solving for mass compared to using a three-equation system involving initial and final momentum and kinetic energy changes. The discussion highlights its utility in physics, particularly in scenarios where only momentum and kinetic energy differences are known. Overall, the equation serves as a practical tool in classical mechanics.
Superstring
Messages
120
Reaction score
0
p^2=p_0^2+2m \Delta k

I derived it using the following:

k=\frac{1}{2}mv^2

p=mv

\frac{dk}{dv}=mv=p

\frac{dp}{dv}=m

\frac{dk}{p}=dv

\frac{dp}{m}=dv

\frac{dk}{p}=\frac{dp}{m}

mdk=pdp

\int_{k_0}^{k}mdk=\int_{p_0}^{p}pdp

m\Delta k=\frac{1}{2}(p^2-p_0^2)

p^2=p_0^2+2m \Delta k

As far as I can tell it's correct, but I though I'd get a second opinion.
 
Last edited:
Physics news on Phys.org
Since p2 = 2mk, where k = ½mv2

mk = ½p2

m (k2 - k1) = ½(p22 - p12)

Bob S
 
Isn't it easier to just note this?

\Delta k = k - k_0 = \frac{p^2}{2m} - \frac{p_0^2}{2m}

Edit: Ninja'd.
 
As well as what Bob S mentioned, I would say your equation is dimensionally valid.
By the way, why you need such an equation? Or what is it mainly used for?
 
It's correct for speeds much smaller than that of light.
 
Thanks for the responses :).

kiwakwok said:
By the way, why you need such an equation? Or what is it mainly used for?

As far as uses, it's a great deal simpler to solve for the mass than if you were to use a 3-equation system with p=mv, p0=mv0, and Δk=(m/2)(v2-v02) if you were only given p, p0 and Δk.
 
So I know that electrons are fundamental, there's no 'material' that makes them up, it's like talking about a colour itself rather than a car or a flower. Now protons and neutrons and quarks and whatever other stuff is there fundamentally, I want someone to kind of teach me these, I have a lot of questions that books might not give the answer in the way I understand. Thanks
Back
Top