Hong Life CalculatorsUnderstanding the Newton's third law calculator — concept and principle
This calculator applies Newton's third law to two bodies pushing on each other: it gives the equal and opposite forces, each body's acceleration, their speeds after a push of given duration and the impulse. It is used in physics classes for examples like two skaters pushing apart, a person stepping off a boat or a gun's recoil.
How it is calculated
F_on B from A = F; F_on A from B = −F
a_A = F ÷ m_A; a_B = F ÷ m_B; a_A ÷ a_B = m_B ÷ m_A
After time t from rest: v = F t ÷ m
Impulse on each body = F × t (equal and opposite)
- No friction is assumed; the force is constant during the contact time.
The principle
Forces always come in pairs. When A pushes B, B pushes A back with a force of the same size in the opposite direction. The two forces act on different bodies, so they do not cancel; each body accelerates according to its own mass. The lighter body gets the larger acceleration and speed. Because the impulses are equal and opposite, the total momentum of the pair stays zero if they start at rest: m_A v_A = m_B v_B.
Worked example
Default inputs: force 240 N, m_A = 60 kg, m_B = 80 kg, contact time 0.5 s.
- Force on B = 240 N; force on A = −240 N.
- a_A = 240 ÷ 60 = 4 m/s²; a_B = 240 ÷ 80 = 3 m/s²; ratio = 1.333.
- v_A = 4 × 0.5 = 2 m/s; v_B = 3 × 0.5 = 1.5 m/s, in opposite directions.
- Impulse on each = 240 × 0.5 = 120 N·s; momentum check: 60 × 2 = 80 × 1.5 = 120.
| Push or pull force between the bodies (N) | Force on body B from A (N) |
|---|---|
| 60 | 60 |
| 120 | 120 |
| 180 | 180 |
| 240 | 240 |
| 300 | 300 |
| 360 | 360 |
| 480 | 480 |
How to read the result
The heavier body moves off more slowly, but both carry the same momentum. This is why a rifle recoils gently compared with the bullet's speed, and why a rocket accelerates by pushing exhaust backwards. Friction with the ground would reduce the speeds in a real situation.
Common mistakes
- Thinking the two forces cancel because they are equal and opposite. They act on different bodies.
- Believing the heavier body pushes harder.
- Forgetting directions when adding velocities.
Frequently asked questions
If forces are equal, why does the lighter body move faster?
Acceleration is force divided by mass, so less mass means more acceleration.
Does the third law apply to gravity?
Yes. You pull the Earth up with the same force it pulls you down.
What is impulse?
Force times time, equal to the change in momentum.
Why does momentum stay zero?
The impulses on the two bodies are equal and opposite.
Sources: Newton's second and third laws and the impulse–momentum theorem (standard introductory mechanics, e.g. OpenStax University Physics Volume 1, chapters 5 and 9)