Question

A pendulum is initially displaced to a height 1 y above the table
top, as in figure 1. It is released with an initial speed of 1 v = 0 . At
the lowest point of its motion, it is at a height 2 y above the
table top, and with a speed of 2 v . Derive an equation of 2 v , in
terms of 1 y and 2 y , assuming that its total mechanical energy is
conserved.

Answer 1

Can you double check if figure 1 looks like this:
http://img823.imageshack.us/img823/6783/1282485265.png

If this is the case, then
equate energies, namely potential and kinetic energies:
Change in potential energy, ΔEp=mg(y2-y1)
Change in kinetic energy, ΔEk=(1/2)m(v2²-v1²)

By the law of conservation of energy, we have
ΔEp+ΔEk=0
Substitute and solve for v2, noting that v1=0.