Energy Formula Audits

Work-Energy Speed Audit

push distance becomes speed

Three constant-force pushes close the loop from work input to final kinetic energy.

mechanics With the force and mass fixed, the push distance sets the work, and the final speed must make exactly the same kinetic energy.

Example

Three constant-force pushes close the loop from work input to final kinetic energy. 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 work against final speed

Start from rest and push in the direction of motion with the same force each time. Work added by the push must equal the final kinetic energy.

Fd=ΔKE=12 m v2F d = \Delta KE = \tfrac{1}{2}\,m\,v^{2}

First push closes

The 4 N push over 4 m adds 16 J. That is exactly the kinetic energy of 4 m/s.

Fd=4 N ⋅ 4 m=16 J=12 2 kg (4 m/s)2=16 JFd = 4\ \text{N}\,\cdot\,4\ \text{m} = 16\ \text{J} = \tfrac{1}{2}\,2\ \text{kg}\,\left(4\ \text{m}/\text{s}\right)^{2} = \hl{16}\ \text{J}
Short pushA force arrow pushes the cart; the final speed is shown by the velocity arrow.mvF

Second push closes

Over 9 m, the same force adds 36 J. The final speed 6 m/s gives the matching kinetic energy.

Fd=4 N ⋅ 9 m=36 J=12 2 kg (6 m/s)2=36 JFd = 4\ \text{N}\,\cdot\,9\ \text{m} = 36\ \text{J} = \tfrac{1}{2}\,2\ \text{kg}\,\left(6\ \text{m}/\text{s}\right)^{2} = \hl{36}\ \text{J}
Middle pushThe same force acts over a longer path, ending with a longer velocity arrow.mvF

Third push closes

Over 16 m, the same force adds 64 J. The final speed 8 m/s closes the largest row.

Fd=4 N ⋅ 16 m=64 J=12 2 kg (8 m/s)2=64 JFd = 4\ \text{N}\,\cdot\,16\ \text{m} = 64\ \text{J} = \tfrac{1}{2}\,2\ \text{kg}\,\left(8\ \text{m}/\text{s}\right)^{2} = \hl{64}\ \text{J}
Long pushThe same force acts over the longest path, ending with the longest velocity arrow.mvF

Work and kinetic energy agree row by row

The distance column controls the work because the force is fixed. The speed column is accepted only when its kinetic energy equals that work.

dWvKE4 m16 J4 m/s16 J9 m36 J6 m/s36 J16 m64 J8 m/s64 J\begin{array}{c|c|c|c}d & W & v & KE \\ \hline 4\ \text{m} & 16\ \text{J} & 4\ \text{m}/\text{s} & 16\ \text{J} \\ 9\ \text{m} & 36\ \text{J} & 6\ \text{m}/\text{s} & 36\ \text{J} \\ 16\ \text{m} & 64\ \text{J} & 8\ \text{m}/\text{s} & 64\ \text{J}\end{array}