An 80.0 kg student running at 3.5 m/s grabs a rope that is hanging vertically. How high will the student swing?

Ek₀ and Ep₀ are initial kinetic and potential energies respectively.

Ek and Ep are final kinetic and potential energies respectively.

Energy in the system is conserved.

Ek₀ + Ep₀ = Ek + Ep

Assume student is one some ground ground, so Ep₀ = 0

At the apex of student's swing, Ek = 0

Ek₀ + 0 = 0 + Ep

Ek₀ = Ep

mv₀²/2 = mgh

Mass cancels out

v₀²/2 = gh

h = v₀² / (2g)

h = (3.5 m/s) ² / (2 * 9.81 m/s²)

h = 0.62 m

He will swing 0.62 m high (relative to where he was running before)