Mastering Equations: How to Graph x against t with Ease

  • Thread starter romd
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In summary, the conversation is about manipulating the equation x(t)=e^(-t/tau) sin(omega*t) into a straight line graph (y=mx+c) of position (x) against time (t). The equation describes a damped oscillator and making coordinate transformations does not seem to be a viable solution. Instead, the function can be plotted as y = me^(-t/tau) sin(omega*t) + b to find the corresponding y value on the line y=mx+b.
  • #1
romd
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Hi, I need help manipulating the equation

x(t)=e^(-t/tau) sin(omega*t)

into a straight line graph (y=mx+c) of position (x) against time (t)

Thanks in advance!

edit: this might have been better off in the homework/coursework section, sorry
 
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  • #2
Do you mean x = m.t + c ?
Are you allowed to make co-ordinate transformations ? The equation describes a damped oscillator so there doesn't seem to be any other way.
 
  • #3
Thanks for the reply!
Probably, I just know I need a straight line out of it.. I think its a little beyond my knowledge, given my college teacher has no idea either. Once I have the equation I'll be set.
 
  • #4
The question is "make a straight line out of what"? The graph of the function you give, [tex]x(t)=e^{-t/\tau} sin(\omega t)[/tex], in the "tx-plane" is certainly not a straight line and there is no y. The graph of that would be a "wave" getting smaller and smaller as t increases. Sometimes you can make a function into a linear graph by plotting the logarithms of the values rather than the values themselves, but that won't work with that "sin" there.

Perhaps you mean that, given this x value, as a function of t, find the corresponding y value on the line y= mx+ b. That's easy- just substitute that function for x:
[itex]y= me^{-t/\tau} sin(\omega t)+ b[/itex]. The values given by those functions will give the position, at time t, of an object moving along the line y= mx+ b.
 

1. What is the purpose of graphing x against t?

The purpose of graphing x against t is to visually represent the relationship between two variables, x and t. By plotting the values of x on the x-axis and the values of t on the y-axis, we can see how changes in one variable affect the other.

2. What are some common mistakes to avoid when graphing x against t?

Some common mistakes to avoid when graphing x against t include using the wrong scale on the axes, not labeling the axes correctly, and not plotting all the data points accurately. It is also important to remember to include units and a title for the graph.

3. How do I determine the slope of a line on a graph of x against t?

To determine the slope of a line on a graph of x against t, you can use the slope formula: slope = (change in y)/(change in x). This means you divide the change in the y-values by the change in the x-values between any two points on the line.

4. Can I use a graph of x against t to make predictions?

Yes, you can use a graph of x against t to make predictions. If the data points form a straight line, you can use the trend of the line to make predictions about future values. However, it is important to note that predictions based on a graph are only as accurate as the data points used to create it.

5. How can I use a graph of x against t to solve equations?

A graph of x against t can be used to solve equations by visually representing the solutions. The x-intercept of the graph represents the value of x when t is equal to 0, and the y-intercept represents the value of t when x is equal to 0. By finding these intercepts, you can solve for either x or t in the equation.

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