The physics
These problems happen in two stages. Treat each stage with its own rule.
Stage 1, the hit. It is over in a flash. No outside push has time to act. So momentum is kept. KE is kept only in an elastic hit. When a bullet sticks, most of its KE turns into heat.
they stick: m u = (m + M) V elastic, target at rest: v₂ = 2m₁u / (m₁ + m₂)
Stage 2, after the hit. A swing keeps energy, so KE becomes PE. A slide on a rough floor loses its KE to friction. A flight off a post is a projectile. Bodies that leave level from one height fall for the same time.
swing: V = √(2gh) slide: s = V² / (2μg) fall: t = √(2H / g)
A bob on a string goes right round only if it moves fast enough. At the top, the string can only pull. So it needs v² ≥ gL there, and V ≥ √(5gL) at the bottom.
Work backwards from what you see. The rise, slide or landing gives the speed after the hit. Momentum then gives the speed before it.