Newton's third law calculator
Solve or verify a one-dimensional action – reaction relationship using m₁a₁ = – m₂a₂, then inspect the equal-and-opposite force pair.
Interaction components
Enter all four to verify, or leave exactly one blank to solve it.Live results
Action and reaction forces – direct check
Use this independent force-only path when you already know the action force and only need its reaction pair.
How to use this Newton's third law calculator
What this calculator does
This calculator checks or solves the one-dimensional relationship m₁a₁ = – m₂a₂. It combines Newton's second-law expression F = ma with Newton's third law, so the interaction force on one object is equal in magnitude and opposite in direction to the interaction force on the other. It is useful for idealized contact, push-off, recoil, collision-force, or classroom exercises where the stated accelerations represent the same interaction pair. It does not replace a full free-body diagram, and it does not claim that the two objects must have equal accelerations: different masses can have different acceleration magnitudes while the interaction forces remain equal and opposite.
When to use it
- Check a mechanics homework solution in which two interacting objects have known masses and accelerations.
- Solve for one missing mass or acceleration when the other three component values are known.
- Convert the same calculation between kilograms, grams, pounds, metres per second squared, feet per second squared, and standard gravity.
- Use the direct force check when an action force is known and you only need the corresponding reaction force.
How to calculate
- The calculator opens with a complete demonstration: 90 kg at +20 m/s² interacting with 30 kg at – 60 m/s². The first screen is already calculated, and Download Excel is immediately available for those values.
- Replace the values in Mass of object 1, Acceleration of object 1, Mass of object 2, and Acceleration of object 2. Enter all four to verify the identity, or leave exactly one of the four blank to solve for that missing component.
- Use each adjacent unit selector to choose the unit for that field. When you change a unit, the calculator converts the current value so the physical quantity stays the same.
- Read Third-law check, Component action force, Component reaction force, Solved component, and Force-pair magnitude. The result updates live as the controls change.
- Optionally open Action and reaction forces – direct check. Enter Action force (direct check) and choose its unit to obtain Direct reaction force.
- Use Download Excel to export the current validated model as a real XLSX workbook. Reset clears the demonstration values and calculated state; after Reset, Excel export is disabled until you enter a complete valid component relationship or a valid direct force.
Input guide
Mass of object 1 and Mass of object 2 are positive decimal masses. Mass 1 unit and Mass 2 unit accept kg, g, or lb. Both masses may be supplied, or one mass may be left blank when the other three component values uniquely determine it. A realistic example is 90 kg and 30 kg. Increasing one mass while holding the other mass and the paired interaction force fixed reduces that object's acceleration magnitude. Negative or zero masses are rejected.
Acceleration of object 1 and Acceleration of object 2 are signed decimal accelerations. Acceleration 1 unit and Acceleration 2 unit accept m/s², ft/s², or standard gravity g₀. A positive sign means one chosen axis direction and a negative sign means the opposite. The demonstration uses +20 m/s² and – 60 m/s². Do not treat the minus sign as “smaller”; it represents direction. Scientific notation and decimal-comma input are intentionally rejected so pasted values are not silently reinterpreted.
Action force (direct check) is an optional signed decimal force, with N or lbf selectable through Direct force unit. The example value is +1800 N. This path is independent of the mass – acceleration solver: changing it updates only the direct force pair. Its sign carries direction, and Direct reaction force is exactly the negative of the entered action force.
Output guide
Third-law check reports whether the four component values satisfy the identity, whether one component was solved, or whether more information is needed. Component action force is m₁a₁ in newtons; Component reaction force is its exact negative. Solved component identifies the field calculated when exactly one component is blank. Force-pair magnitude is the absolute force size, so it is always nonnegative even though the directional forces carry opposite signs. The header pills repeat the relationship state, force magnitude, and number of supplied components as compact status summaries. Direct reaction force is the force-only result from the optional direct-check input.
For the underlying concept, OpenStax's Newton's third law section emphasizes two points that matter when interpreting these results: the forces are equal and opposite, and they act on different bodies. That is why a third-law pair should not be canceled on one object's free-body diagram.
Formula and interpretation
For a one-dimensional interaction pair, write the force exerted by object 1 on object 2 as F₁₂ and the force exerted by object 2 on object 1 as F₂₁. Newton's third law gives F₁₂ = – F₂₁. If the acceleration values supplied here correspond to those interaction forces, Newton's second law gives m₁a₁ = – m₂a₂. The minus sign is directional, not a statement that one force has a smaller magnitude.
The most common conceptual mistake is to say the two forces “cancel.” They do not cancel on either individual object because each force acts on a different object. NASA's overview of Newton's laws of motion describes the same action – reaction principle in the context of interacting bodies and propulsion.
Units also matter. The calculator converts all component values to SI internally, performs the calculation in kilograms, metres per second squared, and newtons, then converts solved values back into the unit currently selected for the field. For standardized SI definitions and unit relationships, consult the NIST SI Units guidance.
Practical limits
- The component equation assumes the accelerations you enter are attributable to the same interaction pair. If gravity, friction, drag, tension, or other external forces also contribute, use a complete free-body model before applying this relationship.
- If all four component values are present but inconsistent, the calculator does not silently overwrite one of them. Clear the component you want to solve, or correct the inconsistent value.
- If an acceleration needed to solve a missing mass is zero, the mass may be indeterminate rather than uniquely solvable. The calculator reports that condition instead of dividing by zero.