Motion and Force Relation Scans

Mass-Force Acceleration Three-Row Scan

mass needs force

Three Newton's-second-law rows hold acceleration fixed while mass changes, exposing force as the scaled output.

Example

With acceleration fixed, Newton's second law makes force scale with mass. 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 mass

Newton's second law is a scaling rule: more mass needs more force for the same acceleration.

F=maF = m a

Light cart

Mass 1 kg at the fixed acceleration needs 3 N.

F=ma=1 kg ⋅ 3 m/s2=3 NF = m a = 1\ \text{kg}\,\cdot\,3\ \text{m}/\text{s}^{2} = 3\ \text{N}
Cart relation rowA cart on level ground carries only the arrows used by this row.aF

Middle cart

Mass 2 kg at the same acceleration needs 6 N.

F=ma=2 kg ⋅ 3 m/s2=6 NF = m a = 2\ \text{kg}\,\cdot\,3\ \text{m}/\text{s}^{2} = 6\ \text{N}
Cart relation rowA cart on level ground carries only the arrows used by this row.aF

Heavy cart

Mass 3 kg at the same acceleration needs 9 N.

F=ma=3 kg ⋅ 3 m/s2=9 NF = m a = 3\ \text{kg}\,\cdot\,3\ \text{m}/\text{s}^{2} = 9\ \text{N}
Cart relation rowA cart on level ground carries only the arrows used by this row.aF

The force column follows mass

Acceleration is held fixed, so doubling mass doubles the required net force; tripling mass triples it.

maF1 kg3 m/s23 N2 kg3 m/s26 N3 kg3 m/s29 N\begin{array}{c|c|c}m & a & F \\ \hline 1\ \text{kg} & 3\ \text{m}/\text{s}^{2} & 3\ \text{N} \\ 2\ \text{kg} & 3\ \text{m}/\text{s}^{2} & 6\ \text{N} \\ 3\ \text{kg} & 3\ \text{m}/\text{s}^{2} & 9\ \text{N}\end{array}
mechanics At 3 m/s squared, masses 1 kg, 2 kg, and 3 kg require net forces 3 N, 6 N, and 9 N.