Show that on a roller coaster with a circular vertical loop the difference in your apparent weight at the top?

Question

Show that on a roller coaster with a circular vertical loop, the difference in your apparent weight at the top of the loop and the bottom of the loop is 6 g's - that is, six times your weight. Ignore friction. Show also that as long as your speed is above the minimum needed, this answer doesn't depend on the size of the loop or how fast you got through it. Beginning height of the coaster is,h, and radius of loop is R.

Answer 1

Acceleration at top
A1=v1^2/r - g
Acceleration bottom
A2=v2^2/r + g

A2 - A1 = (v2^2 - v1^2)/r +2g

Energy at top = Et= KE + PE
=mv1^2/2 + mgh
h=2r
therefore
Et= mv1^2/2 + mg2r

Ebottom=Eb=mv2^2/2
but Eb=Et, therefore
Eb=mv2^2=mv1^2/2 + mg2r
therefore
(mv2^2 -mv1^2)/2 = mg2r
v2^2 -v1^2 = 4*gr
(v2^2 -v1^2)/r = 4*g

Now remember our equation from above

A2 - A1 = (v2^2 - v1^2)/r +2g

sub in (v2^2 -v1^2)/r = 4*g

A2 - A1 = 4g +2g
=6g