Question

How is this formula is derived,
f(n) = 2^{(n-49)/12} * 440Hz ; it is used to find the frequency of any key in Grand piano and will this formula work in Upright piano.

Answer 1

The basic formula for the frequencies of the notes of the equal tempered scale is given by
fn = f0 * (a)n
where
f0 = the frequency of one fixed note which must be defined. A common choice is setting the A above middle C (A4) at f0 = 440 Hz.
n = the number of half steps away from the fixed note you are. If you are on a higher note, n is positive. If you are on a lower note, n is negative.
fn = the frequency of the note n half steps away.
a = (2)1/12 = the twelfth root of 2 = the number which when multiplied by itself 12 times equals 2 = 1.059463094359...

The wavelength of the sound for the notes is found from
Wn = c/fn
where W is the wavelength and c is the speed of sound. The speed of sound depends on temperature, but is approximately 345 m/s at "room temperature."

So,

The following equation gives the frequency f of the nth key.

Answer 2

F=m.a  (“Force = mass x acceleration” - which is Newton’s second law) - and that the velocity of an object is it’s acceleration multiplied by the elapsed time since it started moving (v=a.t).

If I want to know how much time it takes for an object of a given mass to reach some specified speed when acted on by a particular force - then I can “derive an equation” for that by rearranging v=a.t into t=v/a - and rearranging F=m.a into a=F/m - and then substituting the ‘a’ in t=v/a and we get:

t=v/(F/m)

…or…

t=v.m/F

There! We just “derived an formula for the time it takes for a mass ‘m’ to reach a speed of ‘v’ when acted upon by a constant force ‘F’!