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

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

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meowchow said:
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.
 


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
 


meowchow said:
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.
 


Ohh thank you so much! ^____^