Please Help! Science Nerds Needed!!!

[Martin]

Here's another one:

Which decays faster: an Isotope with a short or with a long Half-Life?
I would right away say short because of logic, but it seems to easy to be that, which one is it?

Short.

This can be modeled with a simple differential equation:

(dE/dt) = (k*E), which after integration through separation of variables yields E(t) = E(0)*exp^(k*t) where E(t) is the mass of the isotope present at any given time (I chose E since I use (I) to represent current in a circuit, which can cause confusion). k is used to represent the rate at which the isotope decays, and in this case it would be negative. (exp = e = ~2.71828.... etc., t = time in years)

Let's model this carefully: (Let's assume one isotope has a half-life of 250 years and another has an HL of 500 years), and that you start off with 100 g of that isotope:

50 = 100 exp^(k*250)
50 = 100 exp^(k*500)

Solve for k on both equations. k ends up being -ln(2) / 250 and -ln(2)/500 respectively, or -0.002773 and -0.001386 respectively. The latter has a slower decay rate, and corresponds to a longer half-life, thus the former has a faster decay rate and a shorter half-life.

Hope this in-depth proof was helpful.

 

[john3]

wow you guys get to skip ahead? the school were i go to, im in grade 10, im not sure if we can skip ahead, but if you have prior knowledge of a subject, you take an exam then you get the credit, but in elementar school you dont get skipped.

 

[Martin]

and math, well calculus 3 was particularly challenging and so were some chapters in physics. But again, you are not in college, so you don't have to worry about this stuff (yet).

njoy high school, you'll miss it!

I bet anger boils up when I say sequences and series (as well as Taylor and Maclaurin series). That's usually the part that kills everyone. But unfortunately that stuff may come in handy later for me (especially when I work on Fast Fourier Transforms in DSPs). That, and I absolutely hated chemistry as well.

 

[ncjimn]

I bet anger boils up when I say sequences and series (as well as Taylor and Maclaurin series). That's usually the part that kills everyone. But unfortunately that stuff may come in handy later for me (especially when I work on Fast Fourier Transforms in DSPs). That, and I absolutely hated chemistry as well.

oh yeah, infinite series...fun stuff, that was last semester though. i did good in the class, but at this point (one semester later) i barely remember the majority of what i "learned." we're now working with multiple and line integrals.

i decided not to take differential equations next semester. i'll leave that for the semester after. i need a break from math

and I agree, stuff that seems kind of useless now might come in handy later. talk about half-angle formulas!