# Relativistic and non-relativisitic kinetic energy

• lola2000
In summary, to find the relative difference of 5% between the expressions for relativistic and non-relativistic kinetic energy, the equation 0.05 = [(gamma-1)mc^2 - 0.5mc^2] / 0.5mv^2 can be simplified to 0.525v^2/c^2 = gamma - 1. To solve for v, add 1 to both sides, square both sides, rename v^2/c^2 = X, solve the quadratic equation for X, and then take the square root of X to find the speed at which the expressions vary by 5%.
lola2000

## Homework Statement

at what speed does the expression for relativistic kinetic energy vary from the non-relativistic expression by 5%?

## Homework Equations

Relativistic kinetic energy K=(gamma-1)mc^2
Non-relativistic kinetic energy K=0.5mv^2

## The Attempt at a Solution

I'm not sure how to attempt this!
Should I be finding the difference? ie K(relativistic)-K(non-relativistic)=0.05??

Hi lola2000, welcome to PF!

lola2000 said:
Should I be finding the difference? ie K(relativistic)-K(non-relativistic)=0.05??

Close, you should be finding the relative difference |Krel-Knon-rel|/Knon-rel=0.05 ... make sense?

I see, that makes sense.

But when I do this I get
0.05 = [(gamma-1)mc^2 - 0.5mc^2] / 0.5mv^2
which simplifies to
0.6mv^2 = (gamma-1)mc^2
0.6v^2/c^2 +1 = gamma

which is really nasty to solve! Is there a trick I have missed?

lola2000 said:
I see, that makes sense.

But when I do this I get
0.05 = [(gamma-1)mc^2 - 0.5mc^2] / 0.5mv^2

I assume this is a typo?

which simplifies to
0.6mv^2 = (gamma-1)mc^2

Really?

You are right!
That was a typo - it should have been -0.5mv^2

But I am still stuck with the algebra

I have 0.525v^2/c^2 = gamma - 1

How do I rearrange this??

It is not simplifying!

lola2000 said:
You are right!
That was a typo - it should have been -0.5mv^2

But I am still stuck with the algebra

I have 0.525v^2/c^2 = gamma - 1

How do I rearrange this??

It is not simplifying!

I did not check the previous steps so I cannot guarantee this is the correct equation. But assuming it is, you just need to add 1 to both sides (to have gamma isolated). Then square both sides and rename $$v^2/c^2 = X$$ . Then you will have a quadratic equation for X. Solve, keep only the positive root. That will be $$v^2/c^2$$. Then the answer is the square root of X.

## What is the difference between relativistic and non-relativistic kinetic energy?

Relativistic kinetic energy takes into account the effects of special relativity, such as time dilation and length contraction, on an object's velocity and mass. Non-relativistic kinetic energy ignores these effects and only considers the object's mass and velocity in classical mechanics.

## How is relativistic kinetic energy calculated?

The formula for relativistic kinetic energy is K = (γ - 1)mc², where γ is the Lorentz factor and m is the object's rest mass. This formula takes into account the increasing mass of an object as it approaches the speed of light.

## Can an object have negative relativistic kinetic energy?

No, relativistic kinetic energy is always positive. This is because the Lorentz factor is always greater than or equal to 1, and the object's rest mass is always positive.

## What is the relationship between relativistic and non-relativistic kinetic energy?

As an object's velocity approaches the speed of light, the difference between relativistic and non-relativistic kinetic energy becomes more significant. In the non-relativistic limit (when the object's velocity is much less than the speed of light), the two forms of kinetic energy are nearly equivalent.

## How does relativistic kinetic energy affect an object's momentum?

Relativistic kinetic energy is directly related to an object's momentum through the equation K = (pc)²/2m, where p is the object's momentum and c is the speed of light. As an object's velocity approaches the speed of light, its momentum and relativistic kinetic energy both increase significantly.

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