Motion and Force Relation Scans

Momentum-Speed Three-Row Scan

velocity carries momentum

Three momentum rows hold mass fixed while velocity changes, so momentum follows the velocity column exactly.

Example

With mass fixed, momentum follows velocity exactly. 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 mass fixed and scan velocity

Momentum keeps the mass and velocity together. With fixed mass, larger velocity means larger momentum in the same direction.

p=mvp = m v

Slow cart

At 2 m/s, the momentum is 4 kg*m/s.

p=mv=2 kg ⋅ 2 m/s=4 kg m/sp = m v = 2\ \text{kg}\,\cdot\,2\ \text{m}/\text{s} = 4\ \text{kg}\,\text{m}/\text{s}
Cart relation rowA cart on level ground carries only the arrows used by this row.v

Middle cart

At 4 m/s, the momentum is 8 kg*m/s.

p=mv=2 kg ⋅ 4 m/s=8 kg m/sp = m v = 2\ \text{kg}\,\cdot\,4\ \text{m}/\text{s} = 8\ \text{kg}\,\text{m}/\text{s}
Cart relation rowA cart on level ground carries only the arrows used by this row.v

Fast cart

At 6 m/s, the momentum is 12 kg*m/s.

p=mv=2 kg ⋅ 6 m/s=12 kg m/sp = m v = 2\ \text{kg}\,\cdot\,6\ \text{m}/\text{s} = 12\ \text{kg}\,\text{m}/\text{s}
Cart relation rowA cart on level ground carries only the arrows used by this row.v

Velocity controls momentum when mass is fixed

The mass column stays fixed. The velocity arrows get longer, and the momentum column follows exactly.

mvp2 kg2 m/s4 kg m/s2 kg4 m/s8 kg m/s2 kg6 m/s12 kg m/s\begin{array}{c|c|c}m & v & p \\ \hline 2\ \text{kg} & 2\ \text{m}/\text{s} & 4\ \text{kg}\,\text{m}/\text{s} \\ 2\ \text{kg} & 4\ \text{m}/\text{s} & 8\ \text{kg}\,\text{m}/\text{s} \\ 2\ \text{kg} & 6\ \text{m}/\text{s} & 12\ \text{kg}\,\text{m}/\text{s}\end{array}
mechanics With a 2 kg cart, speeds 2 m/s, 4 m/s, and 6 m/s give momenta 4 kg*m/s, 8 kg*m/s, and 12 kg*m/s.