Motion and Force Relation Scans

Acceleration-Time Speed Three-Row Scan

time adds velocity

Three acceleration rows keep acceleration fixed and scan the time interval that builds the velocity change.

Example

A fixed acceleration adds larger velocity changes when it acts longer. 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 acceleration fixed and scan time

A steady acceleration adds velocity every second. Here the acceleration stays 2 metres per second squared, while the time row changes.

Δv=at\Delta v = a t

First interval

Over 2 s, the velocity gain is 4 m/s.

Δv=at=2 m/s2 ⋅ 2 s=4 m/s\Delta v = a t = 2\ \text{m}/\text{s}^{2}\,\cdot\,2\ \text{s} = 4\ \text{m}/\text{s}
Cart relation rowA cart on level ground carries only the arrows used by this row.va

Second interval

Over 4 s, the velocity gain is 8 m/s.

Δv=at=2 m/s2 ⋅ 4 s=8 m/s\Delta v = a t = 2\ \text{m}/\text{s}^{2}\,\cdot\,4\ \text{s} = 8\ \text{m}/\text{s}
Cart relation rowA cart on level ground carries only the arrows used by this row.va

Third interval

Over 6 s, the velocity gain is 12 m/s.

Δv=at=2 m/s2 ⋅ 6 s=12 m/s\Delta v = a t = 2\ \text{m}/\text{s}^{2}\,\cdot\,6\ \text{s} = 12\ \text{m}/\text{s}
Cart relation rowA cart on level ground carries only the arrows used by this row.va

Time controls the velocity change

The acceleration arrow is the same in every row. The velocity-change arrow grows because the acceleration acts for longer.

atΔv2 m/s22 s4 m/s2 m/s24 s8 m/s2 m/s26 s12 m/s\begin{array}{c|c|c}a & t & \Delta v \\ \hline 2\ \text{m}/\text{s}^{2} & 2\ \text{s} & 4\ \text{m}/\text{s} \\ 2\ \text{m}/\text{s}^{2} & 4\ \text{s} & 8\ \text{m}/\text{s} \\ 2\ \text{m}/\text{s}^{2} & 6\ \text{s} & 12\ \text{m}/\text{s}\end{array}
mechanics A steady 2 m/s squared acceleration acting for 2 s, 4 s, and 6 s gives velocity changes of 4 m/s, 8 m/s, and 12 m/s.