Maclaurin Series - Exponentials 2 (Isaac Physics)

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domm1
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https://isaacphysics.org/questions/m...5-6c2b8159b501

Part C (i). I have that ΔE = E(T+t) - E(t), and as E(T+t) =E(t)e^(-γt) , then ΔE = E(t)((e^(-γt)-1). I'm pretty sure the fraction I'm looking for is ΔE/E(t), which would equal e^(-γt)-1 , however this is wrong. Can someone help me with this part of the question please?
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ghostwalker
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(Original post by domm1)
https://isaacphysics.org/questions/m...5-6c2b8159b501

Part C (i). I have that ΔE = E(T+t) - E(t), and as E(T+t) =E(t)e^(-γt) , then ΔE = E(t)((e^(-γt)-1). I'm pretty sure the fraction I'm looking for is ΔE/E(t), which would equal e^(-γt)-1 , however this is wrong. Can someone help me with this part of the question please?
Try \dfrac{E(t)-E(T+t)}{E(t)}.

No idea why they want it that way, but it works.
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domm1
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(Original post by ghostwalker)
Try \dfrac{E(t)-E(T+t)}{E(t)}.

No idea why they want it that way, but it works.
Oh yes now I've got it, weird they wanted it that way though, I thought that generally when you're thinking of a change in something you'd take the initial value from the final (even though the vice versa works perfectly fine), but I guess it's maybe because e^(-γt) is a value <1, so by finding the change by subtracting final from initial we get a positive value for the fractional change in energy which is a bit nicer than a negative? My only thought for why they'd want it that way instead.
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ghostwalker
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(Original post by domm1)
Oh yes now I've got it, weird they wanted it that way though, I thought that generally when you're thinking of a change in something you'd take the initial value from the final (even though the vice versa works perfectly fine), but I guess it's maybe because e^(-γt) is a value <1, so by finding the change by subtracting final from initial we get a positive value for the fractional change in energy which is a bit nicer than a negative? My only thought for why they'd want it that way instead.
Could be that since they are talking about decay, then it's implicit in the question that they want the fractional loss, hence....
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domm1
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(Original post by ghostwalker)
Could be that since they are talking about decay, then it's implicit in the question that they want the fractional loss, hence....
I've done Part B, however I am quite lost how I have not got the correct answer.

I found the Maclaurin series for e^(-γt) which gave '1 - γt + (γ^2(t)^2)/2 ', and so I substituted this into the equation for the fractional loss, 1-e^(-γt), and got an answer of 'γt - (γ^2(t)^2)/2 ', but is incorrect?
Last edited by domm1; 1 week ago
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ghostwalker
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(Original post by domm1)
I've done Part B, however I am quite lost how I have not got the correct answer.

I found the Maclaurin series for e^(-γt) which gave '1 - γt + (γ^2(t)^2)/2 ', and so I substituted this into the equation for the fractional loss, 1-e^(-γt), and got an answer of 'γt - (γ^2(t)^2)/2 ', but is incorrect?
Too accurate. If \gamma T &lt;&lt; 1 then you can ignore terms in T^2.
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domm1
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(Original post by ghostwalker)
Too accurate. If \gamma T &lt;&lt; 1 then you can ignore terms in T^2.
Oh okay I understand, however how comes you still consider the T^1 term? I understand that the T^2 term would be a lot smaller in comparison, but is this the reason for the remaining of the T^1 term? Is it also for the fact that by ignoring the T^1 term as well, you get the approximation for the fractional change equal to zero, which would be less accurate than just considering the T^1 term (as we also know there must be an energy change within the system, or do we not?).
Last edited by domm1; 1 week ago
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ghostwalker
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(Original post by domm1)
Oh okay I understand, however how comes you still consider the T^1 term? I understand that the T^2 term would be a lot smaller in comparison, but is this the reason for the remaining of the T^1 term? Is it also for the fact that by ignoring the T^1 term as well, you get the approximation for the fractional change equal to zero, which would be less accurate than just considering the T^1 term (as we also know there must be an energy change within the system, or do we not?).

If you're interested in what the change is, then dropping the T^1 term and saying it's 0, would be a 100% error relative to the true value. As an approximation, it leaves a great deal to be desired
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