Deriving Equations (ALL)

Question 1

how do you derive gravitational energy formula? GPE = mgΔh?

Answer 1

energy = force x distance
the force is the force due to gravity and the distance is the change in height therefore…
E = mg x h or GPE = mgΔh

Question 2

how do you derive power = force x velocity?

Answer 2

power = energy / time
power = force x distance / time
distance / time = velocity
therefore P = Fv

Question 3

how do you derive KE = 0.5mv^2

Answer 3

KE = work done = force x distance
KE = ma x d
using suvat v^2 = u^2 + 2as and therefore as = 0.5(v^2 - u^2)
substituting as for ad (as d and s are equal) gives you KE = 0.5m(v^2 - u^2)
if we consider an acceleration from rest, u = 0
therefore KE = 0.5mv^2

Question 4

how do you derive C = 4πɛor?

Answer 4

(because it is a charged sphere you can assume all of its charge is at its centre and treat it like a point charge)
because V = Q /C and V = Q / 4πɛor hence C = 4πɛor

Question 5

how do you derive T^2 = (4π^2 / GM ) x r^3?

Answer 5

F = - GMm / r^2
equate the gravitational force to centripetal force
mrω^2/ r = GMm / r^
substituting ω = 2π / T gets you to...
T^2 = (4π^2 / GM ) x r^3

Question 6

how do you derive 3/2KT = 0.5mc^2bar?

Answer 6

pV = NKT and pV = 1/3Nmc^2bar
NKT = 1/3Nmc^2bar, cancel the Ns
KT = 1/3mc^2bar, times by 3
3KT = mc^2bar and divide by 2 to get...
3/2KT = 0.5mc^2bar