What is the relationship between the equation E^2 - p^2c^2 = m^2c^4

[meowchow]

What is the relationship between the equation E^2 - p^2c^2 = m^2c^4 and E = mc^2?

 

[salvestrom]

What is the relationship between the equation E^2 - p^2c^2 = m^2c^4 and E = mc^2?

p^2c^2 is the momentum component. An object at rest has 0 momentum, reducing the above to E^2=m^2c^4. Taking the square root of both sides leaves you with the more familiar equation, describing the relationship between rest mass and energy.

 

[meowchow]

I understand that thank you but I was wondering how you derived that formula using definitions like e=mc^2γ and p^t=mcγ and p^x=mvγ...this is on our physics midterm tomorrow and my friends and I don't understand how to use these definitions to derive the equation

 

[elfmotat]

I understand that thank you but I was wondering how you derived that formula using definitions like e=mc^2γ and p^t=mcγ and p^x=mvγ...this is on our physics midterm tomorrow and my friends and I don't understand how to use these definitions to derive the equation

Square the energy equation:

E²=m²c⁴/(1-v²/c²)

Simplify:

E²-E²v²/c²=m²c⁴

Replace (Ev)² with (pc²)²:

E²-p²c²=m²c⁴To get the equation Ev=pc²:

E=γmc²
p=γmv

Eliminate γ & m:

Ev=pc^2**This seems like unnecessary algebra. I don't see why you'd have to derive the above equations from other energy/momentum equations. You should also learn about four-vectors, because the first equation actually comes from the magnitude of the four-momentum.

 

[meowchow]

Ohh thank you so much! ^____^