Energy Formula Audits

Speed-Squared Energy Audit

three exact speeds

Three cart speeds are checked against the speed-squared column so kinetic energy cannot be read as a merely linear speed rule.

Example

Kinetic energy is checked across three exact speed cases, with the speed-squared column exposed so the square cannot hide. 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 the square, not just the speed

Keep the mass fixed at 2 kg. The whole speed quantity is squared, so the table must grow by speed squared, not by speed alone.

KE=12 m v2KE = \tfrac{1}{2}\,m\,v^{2}

First exact speed case

At 4 m/s, the speed-squared column is 16. With the fixed mass, the kinetic energy closes at 16 J.

KE=12 2 kg (4 m/s)2=16 J=16 JKE = \tfrac{1}{2}\,2\ \text{kg}\,\left(4\ \text{m}/\text{s}\right)^{2} = 16\ \text{J} = \hl{16}\ \text{J}
Speed case: lowA cart moving with a short velocity arrow.mv

Second exact speed case

At 6 m/s, the speed-squared column is 36. The energy follows that square and closes at 36 J.

KE=12 2 kg (6 m/s)2=36 J=36 JKE = \tfrac{1}{2}\,2\ \text{kg}\,\left(6\ \text{m}/\text{s}\right)^{2} = 36\ \text{J} = \hl{36}\ \text{J}
Speed case: middleA cart moving with a medium velocity arrow.mv

Third exact speed case

At 8 m/s, the speed-squared column is 64. The arrow is twice the first arrow, but the energy is four times the first energy.

KE=12 2 kg (8 m/s)2=64 J=64 JKE = \tfrac{1}{2}\,2\ \text{kg}\,\left(8\ \text{m}/\text{s}\right)^{2} = 64\ \text{J} = \hl{64}\ \text{J}
Speed case: highA cart moving with a long velocity arrow.mv

The three rows expose the square

Scan down the rows. The speed values grow steadily, but the energy tracks the speed-squared column exactly.

vspeed squaredKE4 m/s1616 J6 m/s3636 J8 m/s6464 J\begin{array}{c|c|c}v & \text{speed squared} & KE \\ \hline 4\ \text{m}/\text{s} & 16 & 16\ \text{J} \\ 6\ \text{m}/\text{s} & 36 & 36\ \text{J} \\ 8\ \text{m}/\text{s} & 64 & 64\ \text{J}\end{array}
mechanics With a fixed 2 kg mass, the speed-squared values 16, 36, and 64 close exactly to 16 J, 36 J, and 64 J.