Motion and Force Relation Scans

Velocity-Time Distance Three-Row Scan

time stretches distance

Three constant-velocity rows hold speed fixed while time changes, so distance can be read as a growing ledger.

Example

Holding velocity fixed while time changes makes distance grow 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

Hold velocity fixed and scan time

A constant velocity row is a distance ledger. The speed stays 3 m/s, and each larger time extends the travel distance by the same amount again.

x=vtx = v t

Short time

After 2 s, the cart covers 6 m.

x=vt=3 m/s ⋅ 2 s=6 mx = v t = 3\ \text{m}/\text{s}\,\cdot\,2\ \text{s} = 6\ \text{m}
Cart relation rowA cart on level ground carries only the arrows used by this row.startxv

Middle time

After 4 s, the same velocity covers 12 m.

x=vt=3 m/s ⋅ 4 s=12 mx = v t = 3\ \text{m}/\text{s}\,\cdot\,4\ \text{s} = 12\ \text{m}
Cart relation rowA cart on level ground carries only the arrows used by this row.startxv

Long time

After 6 s, the distance reaches 18 m.

x=vt=3 m/s ⋅ 6 s=18 mx = v t = 3\ \text{m}/\text{s}\,\cdot\,6\ \text{s} = 18\ \text{m}
Cart relation rowA cart on level ground carries only the arrows used by this row.startxv

The distance column grows with time

The velocity column is unchanged. The time column changes, so the distance column changes in the same ratio.

vtx3 m/s2 s6 m3 m/s4 s12 m3 m/s6 s18 m\begin{array}{c|c|c}v & t & x \\ \hline 3\ \text{m}/\text{s} & 2\ \text{s} & 6\ \text{m} \\ 3\ \text{m}/\text{s} & 4\ \text{s} & 12\ \text{m} \\ 3\ \text{m}/\text{s} & 6\ \text{s} & 18\ \text{m}\end{array}
mechanics At 3 m/s, the times 2 s, 4 s, and 6 s give distances 6 m, 12 m, and 18 m.