Newton's Law of Cooling

https://isaacphysics.org/questions/maths_ch7_3_q3?board=4b402709-2770-4070-9347-e32a2dcb45c8&stage=a_level

I completed this question but I don't really understand the second part of part d: "time constant τ of the cooling."

I just guessed it was 1/i but I dont actually know what the 'time constant' means - could someone please explain?

(for first part of part d, I got theta(t) = theta_0*e^(-it) )

Reply 1

mqb2766

Original post by mosaurlodon

https://isaacphysics.org/questions/maths_ch7_3_q3?board=4b402709-2770-4070-9347-e32a2dcb45c8&stage=a_level
I completed this question but I don't really understand the second part of part d: "time constant τ of the cooling."
I just guessed it was 1/i but I dont actually know what the 'time constant' means - could someone please explain?
(for first part of part d, I got theta(t) = theta_0*e^(-it) )

Its similar to a half life, but its the time taken to decay to e^(-1) so ~37% of the inital value. So every time constant the exponential decays by 63% ~2/3. So
e^(-2t)
has a time constant of tau=1/2 and sampling every 1/2 s, it will be e^(-1), e^(-2)=e^(-1)e^(-1), e^(-3)=e^(-1)e^(-2), ... so decaying by e^(-1) every time constant. After 4 or 5 time constants, the exponential will have decayed to ~1% of the initial value and visually this is close to zero so you could use this to define the time range over which you sketched the exponential.

You can write a decaying exponential as
Ae(-t/tau)
and tau is the time constant. For your question tau=1/i as you say. The time constant simply scales (transformation) the time variable as you can write the exponent as -t*(1/tau).

(edited 1 month ago)

Reply 2

mosaurlodon

Thank you

So the time constant is always related to some decaying exponential ae(-b/tau)
and basically means the time taken to decay by the factor of the base^-1 in this context e^-1
so is the 'time constant' a specific term always related to base e or can it be general - the time taken to decay by a factor of any base^-1?

Reply 3

mqb2766

Original post by mosaurlodon

Thank you

Its as in #1, so the time taken to decay to e^(-1) of the initial value.
https://en.wikipedia.org/wiki/Time_constant

You should be working base e, as its the natural thing to do. Anything else is unnatural. If the time constant was dependent on the base, it would be less useful as you can always change base for an exponential and the time constant for a curve wouldnt have a unique value as it would depend on the representation.