How Does Radio Wave Impact on Ionospheric Electron Integrate in Physics?

In summary, the conversation discusses the integration of the equation a=-eE/m, where the focus is on the x axis. The final answer involves integrating twice and including a definite integral, with some confusion about the placement of parentheses.
  • #1
Astrum
269
5
This isn't a HM question, and I'm asking for an explanation.

This is "The effect of a Radio Wave on an Ionospheric Electron"

The integration is weird, I don't follow what is being done.

[tex]a=\frac{-eE}{m}[/tex] - reworking of F=ma

[tex]\frac{-eE}{m}sin(\omega t[/tex]

only interested in the x axis.

[tex]\int\frac{dv}{dt}=\int^{t}_{0}a_{0}sin(\omega t) dt[/tex]

This becomes: [tex]v(t)=v_{0}-\frac{a_{0}}{\omega}cos(\omega t-1)[/tex]
- I don't get where this came from, I understand the indefinite integration, but not where the "ωt-1" came from.

And the last step:

[tex]\int\frac{dx}{dt}=\int^{t}_{0}[v_{0}-\frac{a_{0}}{\omega}cost(\omega t-1)]dt[/tex]

= [tex]x_{0} + (v_{0}+\frac{a_{0}}{\omega})t-\frac{a_{0}}{\omega^{2}}sin(\omega t)[/tex]

Not sure where the final answer comes from. Could't you just integrate it twice, then tack on the definite integral?
 
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  • #2
I think some parentheses are in the wrong place.
$$\int_0^t a_0 \sin(\omega t)\, dt = \left[ -\frac{a_0}{\omega} \cos(\omega t)\right]_0^t$$
$$= - \frac{a_0}{\omega}(\cos(\omega t) - \cos 0)$$
$$= - \frac{a_0}{\omega}(\cos(\omega t) - 1)$$
 

Related to How Does Radio Wave Impact on Ionospheric Electron Integrate in Physics?

1. What is the definition of definite integration?

Definite integration is a mathematical process used to find the area under a curve or the accumulation of a quantity over a specific interval. It is represented by the symbol ∫ and involves finding the antiderivative of a function and evaluating it at the upper and lower bounds of the interval.

2. How is definite integration related to Newton's second law of motion (F=ma)?

Newton's second law of motion states that the force (F) acting on an object is equal to its mass (m) multiplied by its acceleration (a). Definite integration can be used to calculate the total force exerted on an object by finding the area under the curve of its acceleration over a specific interval of time.

3. What are the steps for solving a definite integration problem?

The steps for solving a definite integration problem are as follows:

  1. Identify the function and its bounds.
  2. Find the antiderivative of the function.
  3. Substitute the upper and lower bounds into the antiderivative.
  4. Calculate the difference between the two resulting values to find the definite integral.

4. What are the applications of definite integration in science?

Definite integration has various applications in science, including:

  • Calculating the displacement, velocity, and acceleration of an object in physics.
  • Finding the total mass of a substance in chemistry.
  • Determining the total energy of a system in thermodynamics.
  • Measuring the amount of light absorbed by a material in optics.

5. What is the difference between definite integration and indefinite integration?

The main difference between definite integration and indefinite integration is that definite integration has specific bounds, while indefinite integration does not. Definite integration results in a numerical value, while indefinite integration results in a function with a constant of integration. Additionally, definite integration is used to find total quantities, while indefinite integration is used to find general solutions to differential equations.

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