Newton's Third Law Calculator

Third-law homework looks simple until you are juggling two masses, two accelerations, and a negative sign that keeps flipping on scratch paper. Push a wall and it pushes back—that everyday instinct is Newton's third law—but the worksheet still wants numbers. Rockets expel gas downward and rise upward; skaters push apart when they shove off each other. Think of it like two ice boats linked by a push: whatever momentum one gains, the other takes an equal and opposite shove.

This calculator solves m₁a₁ = −m₂a₂ for the missing mass or acceleration on either object. The formula line below the answer panel stays visible as you work and updates when you change what you are solving for—no need to calculate first just to see which equation applies.

How to Use This Newton's Third Law Calculator

• Select the unknown from the dropdown: acceleration a₂, a₁, mass m₂, or m₁. The output field grays out and shows Calculated.
• Enter the three known values for the two interacting objects. Mix unit systems freely— pounds with m/s², or kilograms with ft/s²—each dropdown converts internally.
• Use consistent sign conventions. If a₁ is positive (right), a₂ should come out negative (left) for a matched action–reaction pair.
• Watch the formula line below the answer panel—it updates with your mode (for example, Acceleration a₂ — m₁·a₁ = −m₂·a₂ when solving for a₂).
• Click Calculate. The missing value appears in the answer panel and in the calculated field.

To get started on a classic recoil-style problem, leave the mode on Calculate a₂, enter both masses and the known acceleration of object 1—the tool reports the opposite-signed acceleration for object 2.

Newton's Third Law Formulas and Practical Applications

Forces always arrive in pairs—equal in magnitude, opposite in direction, acting on different bodies:

F_action = −F_reaction

Substituting F = ma for each object gives the form this calculator uses:

m₁a₁ = −m₂a₂

Rearrange depending on what is unknown. Solving for the second acceleration: a₂ = −m₁a₁ / m₂. Finding the first acceleration: a₁ = −m₂a₂ / m₁. Back-solving mass: m₂ = −m₁a₁ / a₂ and m₁ = −m₂a₂ / a₁. The lighter object accelerates more for the same interaction force—same idea as a small rowboat shooting backward when you throw a heavy anchor forward.

Worked example: finding a₂ from two masses and a₁

Box A (m₁ = 90 kg) accelerates at a₁ = 20 m/s² right while box B (m₂ = 30 kg) responds. Then a₂ = −(90 × 20) / 30 = −60 m/s²—leftward. The interaction force magnitude is |F| = m₁a₁ = 1,800 N on each body, reversed in direction.

Worked example: finding m₂ from known accelerations

A 2 kg cart recoils at −4 m/s² when a 0.5 kg glider accelerates at 10 m/s² on a track. Check: m₂ = −m₁a₁ / a₂ = −(0.5 × 10) / (−4) = 1.25 kg. In our testing, sign errors on a₂ are the most common reason recoil answers come out wrong.

When drawing free-body diagrams, put only the force on each object—not both members of the pair on one sketch. Object 1 feels the force from object 2; object 2 feels the equal and opposite force from object 1. That split is what makes third-law problems click once the algebra is done.

Where third-law pairs show up

• Recoil problems—gun and bullet, skater and wall, rocket and exhaust plume.
• Contact pairs—foot pushing ground during a sprint, hammer striking a nail.
• Lab carts—two coupled carts or a spring-launch setup during collisions unit.
• Intro dynamics homework—problems that name two bodies and ask for the unknown acceleration or mass.

Standard Units and Conversion Tables

Mass units

Acceleration units

Mix lb with m/s² on one form if that matches your data—the solver converts to SI before applying m₁a₁ = −m₂a₂. Moving onto a quick check: if m₁a₁ and −m₂a₂ do not match after rounding, recheck signs on both accelerations.

Frequently Asked Questions

What is Newton's third law of motion?

For every action there is an equal and opposite reaction: F_action = −F_reaction, or m₁a₁ = −m₂a₂ when you write it with Newton's second law for each body.

Do action and reaction forces cancel?

Not when analyzing one object—they act on different bodies. A horse pulls a cart forward; the cart pulls the horse backward with equal force. Each object's motion depends only on forces applied to that object, not the force it applies elsewhere.

What does the negative sign mean in m₁a₁ = −m₂a₂?

Opposite direction. If a₁ is positive (right), a₂ should be negative (left) for a matched interaction pair, and vice versa.

Can I mix unit systems in one calculation?

Yes. Pick units per field from the dropdowns; the tool converts to SI before solving, then displays the answer in the unit you selected for the calculated variable.

How does this differ from the Newton's Second Law Calculator?

This tool pairs two interacting objects through m₁a₁ = −m₂a₂. The Newton's Second Law Calculator focuses on a single body with force, mass, velocity change, and time. For direct F = ma with three variables, use the Force Calculator.

What happens if acceleration is zero when solving for mass?

Division by zero is blocked. Solving for m₂ requires non-zero a₂; solving for m₁ requires non-zero a₁.

Projectile motion after a launch? Try the Trajectory Calculator. For spring restoring forces, open the Hooke's Law Calculator. Browse all physics tools on RapidRatio.