Energy Formula Audits

Drop-Height Speed Audit

height becomes speed

Three frictionless drops balance gravitational potential energy against bottom kinetic energy, row by row.

Example

Three frictionless drops balance gravitational potential energy against bottom kinetic energy, row by row. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Audit height against bottom speed

Use the same mass and gravity each time. The potential energy from height must equal the bottom kinetic energy on the frictionless track.

mgh=12 m v2mgh = \tfrac{1}{2}\,m\,v^{2}

First drop closes

A drop of 5 m stores 100 J. The bottom speed 10 m/s gives the same kinetic energy.

mgh=2 kg ⋅ 10 m/s2 ⋅ 5 m=100 J=12 2 kg (10 m/s)2=100 Jmgh = 2\ \text{kg}\,\cdot\,10\ \text{m}/\text{s}^{2}\,\cdot\,5\ \text{m} = 100\ \text{J} = \tfrac{1}{2}\,2\ \text{kg}\,\left(10\ \text{m}/\text{s}\right)^{2} = \hl{100}\ \text{J}
Drop-height auditA raised mass drops along a frictionless track and leaves the bottom with a velocity arrow.mvh

Second drop closes

A drop of 20 m stores 400 J. The bottom speed 20 m/s closes the same ledger.

mgh=2 kg ⋅ 10 m/s2 ⋅ 20 m=400 J=12 2 kg (20 m/s)2=400 Jmgh = 2\ \text{kg}\,\cdot\,10\ \text{m}/\text{s}^{2}\,\cdot\,20\ \text{m} = 400\ \text{J} = \tfrac{1}{2}\,2\ \text{kg}\,\left(20\ \text{m}/\text{s}\right)^{2} = \hl{400}\ \text{J}
Drop-height auditA raised mass drops along a frictionless track and leaves the bottom with a velocity arrow.mvh

Third drop closes

A drop of 45 m stores 900 J. The bottom speed 30 m/s matches that larger energy.

mgh=2 kg ⋅ 10 m/s2 ⋅ 45 m=900 J=12 2 kg (30 m/s)2=900 Jmgh = 2\ \text{kg}\,\cdot\,10\ \text{m}/\text{s}^{2}\,\cdot\,45\ \text{m} = 900\ \text{J} = \tfrac{1}{2}\,2\ \text{kg}\,\left(30\ \text{m}/\text{s}\right)^{2} = \hl{900}\ \text{J}
Drop-height auditA raised mass drops along a frictionless track and leaves the bottom with a velocity arrow.mvh

Each row balances potential and kinetic energy

The height column makes potential energy. The speed column makes kinetic energy. The two energy columns match row by row.

hPEvKE5 m100 J10 m/s100 J20 m400 J20 m/s400 J45 m900 J30 m/s900 J\begin{array}{c|c|c|c}h & PE & v & KE \\ \hline 5\ \text{m} & 100\ \text{J} & 10\ \text{m}/\text{s} & 100\ \text{J} \\ 20\ \text{m} & 400\ \text{J} & 20\ \text{m}/\text{s} & 400\ \text{J} \\ 45\ \text{m} & 900\ \text{J} & 30\ \text{m}/\text{s} & 900\ \text{J}\end{array}
mechanics The height column creates the energy ledger, and each bottom speed is accepted only when its kinetic energy matches that stored energy.