How to Determine the Asymptotic Approach of v to c Correct to Powers of 1/t^2?

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In summary, the conversation discusses the need to determine the asymptotic approach of v to c, correct to powers of 1/t^2 in an equation involving the speed of light, electrical field, and mass. The speaker considers using a power series expansion and the binomial theorem to solve the problem.
  • #1
mmwave
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I need to "determine the asymptotic approach of v to c, correct to powers of 1/t^2" in the equation below.

[tex]v = \frac{eEt/m_o}{\sqrt{1 + (eEt)^2/(m_oc)^2}}[/tex]

Clearly the asymptote is c (speed of light) and I think I'm being asked to find an expression like constant * (1 + a1 / t + a2 / t^2 + ...) but I have not been able to do so. Power series don't work and binomial thm doesn't apply. Please help.
 
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  • #2
1. Rewrite your expression as:
[tex]v=\frac{c}{\sqrt{1+\epsilon}},\epsilon=\frac{(m_{0}c)^{2}}{(eEt)^{2}}[/tex]
2. Make a power series expansion in [tex]\epsilon[/tex]
 
  • #3
Thanks Arildno. I guess I should have studied calculus in Norway.

Once I followed your advice I could also see that for any particle capable of reaching v approx. equal to c, I can use the binomial theorem.
 

FAQ: How to Determine the Asymptotic Approach of v to c Correct to Powers of 1/t^2?

What is the meaning of asymptotic approach?

Asymptotic approach is a mathematical concept that refers to the behavior of a function or curve as it approaches a certain value or point. It describes the trend of a function as the independent variable gets closer and closer to a particular limit or infinity.

How is asymptotic approach different from convergence?

Asymptotic approach and convergence are related concepts, but they have some key differences. Convergence refers to the idea that a sequence of numbers or values will eventually get closer and closer to a particular value. Asymptotic approach, on the other hand, describes the behavior of a function as it approaches a particular value or point.

What is the significance of determining asymptotic approach?

Determining the asymptotic approach of a function is important in many fields of science, such as physics, engineering, and economics. It helps in understanding the behavior and trends of functions, which can be used to make predictions and solve real-world problems.

How is the asymptotic approach of a function calculated?

The asymptotic approach of a function can be calculated using various mathematical techniques, such as limits, derivatives, and integrals. It is important to note that the method of calculation may vary depending on the type of function and the specific limit or point being approached.

Can the asymptotic approach of a function change?

Yes, the asymptotic approach of a function can change depending on the value or point being approached. For example, a function may approach a certain value at a different rate or behavior compared to another value. Therefore, it is important to specify the limit or point when discussing the asymptotic approach of a function.

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