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How to "derive the leading correction" to an energy expression?

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David Z
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problem question

The kinetic energy of motion of a particle is the relativistic total energy minus the rest energy.

(a) A particle has rest mass $M$ and speed $v$. If $v \ll c$, then show that the kinetic energy due to motion is approximated by the well-known non-relativistic expression for the kinetic energy.

(b) Derive the leading correction to the non-relativistic expression.

In the above problem, I've done part (a) by expanding the Lorentz factor as a power series and then truncating the "neglible""negligible" terms.

However, I'm stumped for deriving the "leading correcton". I've read the Wikipedia article on leading corrections: https://en.wikipedia.org/wiki/Leading-order_term, but I still don't understand what the problem is asking for exactly?

problem question

In the above problem, I've done part (a) by expanding the Lorentz factor as a power series and then truncating the "neglible" terms.

However, I'm stumped for deriving the "leading correcton". I've read the Wikipedia article on leading corrections: https://en.wikipedia.org/wiki/Leading-order_term, but I still don't understand what the problem is asking for exactly?

The kinetic energy of motion of a particle is the relativistic total energy minus the rest energy.

(a) A particle has rest mass $M$ and speed $v$. If $v \ll c$, then show that the kinetic energy due to motion is approximated by the well-known non-relativistic expression for the kinetic energy.

(b) Derive the leading correction to the non-relativistic expression.

In the above problem, I've done part (a) by expanding the Lorentz factor as a power series and then truncating the "negligible" terms.

However, I'm stumped for deriving the "leading correcton". I've read the Wikipedia article on leading corrections: https://en.wikipedia.org/wiki/Leading-order_term, but I still don't understand what the problem is asking for exactly?

Source Link

How to "derive the leading correction" to an expression?

problem question

In the above problem, I've done part (a) by expanding the Lorentz factor as a power series and then truncating the "neglible" terms.

However, I'm stumped for deriving the "leading correcton". I've read the Wikipedia article on leading corrections: https://en.wikipedia.org/wiki/Leading-order_term, but I still don't understand what the problem is asking for exactly?