# Nuclear Chain Reactions

https://isaacphysics.org/questions/maths_ch7_3_q5?board=6edff5fe-5916-4740-b695-bf0a30b59157&stage=a_level

I don't really know what I'm doing wrong with this question, Im assuming it wants me to find the time until the number density decreases by ~63% so I believe this is the right method but I dont know why this is wrong - could someone please take a look?
Help much appreciated.
Working:
Youre correct with your
n = n0 e^((d-1)rt)
Youd expect the solution to be like ~ e^(t/tau) so its simply a case of spotting what tau is
https://en.wikipedia.org/wiki/Exponential_growth#Basic_formula
so the inverse of the term scaling time.
(edited 1 month ago)
Oh ok I managed to get it but im a little confused - in previous isaac phy qs "tau" has been the time constant for a value to reduce by 63% - is this a unique case because the question states f(t/tau​) or does tau have multiple definitions ?

edit: nevermind its exponential growth so it obviously cant decrease by 63% - so in "growth" situations is tau this definition and in "decay" situations is it the previous definition in first paragraph?

Also for the second part which im presuming it asks for the derivative
why isnt dn/dt = n_0 * (d-1) *r
(when t=0) so e^.. ignored
(edited 1 month ago)
Original post by mosaurlodon
Oh ok I managed to get it but im a little confused - in previous isaac phy qs "tau" has been the time constant for a value to reduce by 63% - is this a unique case because the question states f(t/tau​) or does tau have multiple definitions ?
Also for the second part which im presuming it asks for the derivative
why isnt dn/dt = n_0 * (d-1) *r
(when t=0) so e^.. ignored

Usually the time constant is defined in terms of a decaying exponential, so the exponent is negative. So youd have something like
e^(-t/tau)
Then when t=tau, this is 1/e = e^(-1)~0.37 of the initial value or its decayed by about 63%. So every time the time increases by tau, the exponential decays to e^(-1) of the previous value.

Here, the exponent is (probably) positive so a growth exponential so
e^(t/tau)
The time constant is how how it takes to grow to e^1 ~ 2.7 ~ 1/0.37 of the initial value. So every time the time increases by tau, the exponetial grows by e^1 of the previous value. Note were assuming d>1 so the (d-1)r is > 0 so growing.

In both cases its just the reciprocal of the (abs of) the term scaling time and represents how long it takes to decay / grow by e^1
Original post by mosaurlodon
Oh ok I managed to get it but im a little confused - in previous isaac phy qs "tau" has been the time constant for a value to reduce by 63% - is this a unique case because the question states f(t/tau​) or does tau have multiple definitions ?
edit: nevermind its exponential growth so it obviously cant decrease by 63% - so in "growth" situations is tau this definition and in "decay" situations is it the previous definition in first paragraph?
Also for the second part which im presuming it asks for the derivative
why isnt dn/dt = n_0 * (d-1) *r
(when t=0) so e^.. ignored

For b) its simply asking for the solution n(t). It could be clearer.
Ah thank you for all your help! I managed to solve the rest of the parts

For the time constant, I get how in this case it measures the time taken to increase by a factor of e, but you mentioned that for situations with decay, tau is the time taken for the value to reduce by "1/e = e^(-1)~0.37 of the initial value or its decayed by about 63%"

However, I recall doing some question where tau was the time taken to reach 63%, i.e. decay by 37% and there are some articles/resources on google that say it is the time taken to decay by 37% rather than 63% so tbh im still not completely sure on the definition of tau for decay situations.

e.g:
https://saving.em.keysight.com/en/used/knowledge/formulas/time-constant-formula#:~:text=In%20the%20realm%20of%20electronics,value%20following%20a%20step%20input.
"Specifically, it's the time required for a system's response to reach approximately 63.2% of its final value"
Original post by mosaurlodon
Ah thank you for all your help! I managed to solve the rest of the parts
For the time constant, I get how in this case it measures the time taken to increase by a factor of e, but you mentioned that for situations with decay, tau is the time taken for the value to reduce by "1/e = e^(-1)~0.37 of the initial value or its decayed by about 63%"
However, I recall doing some question where tau was the time taken to reach 63%, i.e. decay by 37% and there are some articles/resources on google that say it is the time taken to decay by 37% rather than 63% so tbh im still not completely sure on the definition of tau for decay situations.
e.g:
https://saving.em.keysight.com/en/used/knowledge/formulas/time-constant-formula#:~:text=In%20the%20realm%20of%20electronics,value%20following%20a%20step%20input.
"Specifically, it's the time required for a system's response to reach approximately 63.2% of its final value"

I guess thats a curve like
1 - e^(-t/tau)
so its charging up to 1 from 0, rather than decaying from 1 to 0 if its
e^(-t/tau)
So really its about the time taken for the (basic) exponential to decay to 1/e of the initial or previous value (or growing).
(edited 1 month ago)