At any point along the track, the rollercoaster's mechanical energy is equal to the sum of its potential and kinetic energies,
M = P + K
and we assume the coaster's total mechanical energy is conserved throughout its motion along the track, so that M is the same at every point.
At the first point,
K + 75,000 J = 90,000 J ⇒ K = 15,000 J
At the second point, since M = 90,000 J everywhere,
K + 45,000 J = M ⇒ K = 45,000 J
At the third point,
K + 0 J = M ⇒ K = 90,000 J
The same principles apply to the pendulum, so M = 5,100 J throughout its swing.
At the first point,
K + 4,500 J = 5,100 J ⇒ K = 600 J
At the second point,
K + 0 J = M ⇒ K = 5,100 J
At the third point,
600 J + P = M ⇒ P = 4,500 J
