Newton's Third Law: Pushing Carts

Introduction

Newton's third law states that when two objects interact, the force each exerts on the other is equal in magnitude and opposite in direction. This action-reaction pairing is not a consequence of equilibrium — the two forces act on different objects, and neither cancels the other in any equation of motion. The pushing-carts scenario makes this concrete: a compressed spring between two carts on a frictionless track releases its energy, exerting the same force magnitude on each cart for the same duration. Both carts receive identical impulses, and the total momentum of the system stays at zero from start to finish.

The topic anchors the Newton's-laws curriculum because it cleanly separates two ideas that students often conflate: equal forces and equal accelerations. The sim's p_tot readout demonstrates momentum conservation directly, while the v₁ and v₂ readouts show that speed after the push scales with the inverse of mass. Engineers apply this same relationship whenever a system must transfer momentum internally — from rocket propulsion to automotive crash design to recoil management in firearms.

A common first guess is that if the force is the same on both carts, both carts should move at the same speed afterward. The simulator refutes this: with m₁ = 1 kg and m₂ = 3 kg, the v₁ readout reaches approximately −3.46 m/s while v₂ reaches only about +1.15 m/s — a 3:1 speed ratio that matches the inverse of the 1:3 mass ratio exactly.

The Physics Explained

Newton's third law asserts that for every action force there is a reaction force equal in magnitude and opposite in direction: F₁₂ = −F₂₁. In the cart simulator, the compressed spring pushes cart 1 to the left and cart 2 to the right with the same force at every instant of the push phase. The red force arrows in the simulation are the same length on both carts — a direct visual encoding of this equality. Because the forces act for the same duration, both carts receive the same impulse magnitude J = F · Δt, and the p_tot readout stays at 0.00 kg·m/s throughout the run.

Newton's second law connects those equal forces to the accelerations through F = m · a, rearranged as a = F/m. Because the force is the same but the masses differ, the accelerations differ in proportion to the inverse of the masses: a₁/a₂ = m₂/m₁. With the default slider values of m₁ = 1 kg and m₂ = 3 kg, cart 1 accelerates three times as hard as cart 2 under the identical spring force. After the push phase ends — visible as the disappearance of the red arrows — the v₁ readout shows approximately −3.46 m/s and v₂ shows approximately +1.15 m/s, a ratio of exactly 3:1.

The equal-mass case sharpens the distinction. Setting m₁ = m₂ = 2 kg in the simulator, both carts receive the same impulse and have the same mass, so both reach the same final speed in opposite directions — the v₁ and v₂ readouts show equal magnitudes with opposite signs, and p_tot stays at 0.00 kg·m/s. This is the symmetric version of the third-law pair: the action-reaction forces still exist and are still equal, but now the second-law response is also equal because the inertia is matched.

One important boundary of the model: the simulation treats the spring as delivering its impulse during a brief push phase, then releasing the carts to coast freely. The p_tot readout remains stable through both phases, confirming that no external horizontal force acts on the system. Any small numerical drift visible in p_tot beyond ±0.05 kg·m/s is a sub-step integration artifact; the simulation snaps velocities to the analytical solution at the moment the spring releases, keeping that drift near zero.

Key Equations

The spring exerts force F₁₂ on cart 1 and force F₂₁ on cart 2. These are equal in magnitude and opposite in sign. With the default configuration (m₁ = 1 kg, m₂ = 3 kg), the spring force magnitude is the same on both carts at every instant — the red arrows in the simulator are identical in length throughout the push phase, encoding F₁₂ = −F₂₁ visually.

Rearranged to a = F/m, this relates the shared force to each cart's acceleration. The force magnitude is the same on both carts; the accelerations differ because the masses differ. For a worked example with a 6 N force: cart 1 at 1 kg produces a₁ = 6/1 = 6 m/s², and cart 2 at 2 kg produces a₂ = 6/2 = 3 m/s². The lighter cart accelerates twice as fast under the same force.

Dividing the two second-law expressions cancels the common force and yields the acceleration ratio directly. For m₁ = 1 kg and m₂ = 2 kg: a₁/a₂ = 2/1 = 2. Cart 1 accelerates twice as hard as cart 2. The simulator's v₁ and v₂ readouts reflect this ratio in the final speeds because both carts experience the force for the same duration. For the default slider setting of m₁ = 1 kg and m₂ = 3 kg, the ratio is 3:1, and the readouts confirm v₁ ≈ −3.46 m/s versus v₂ ≈ +1.15 m/s.

Because the action-reaction forces are equal and opposite, the impulses they deliver sum to zero, and the total momentum of the isolated system is conserved at its initial value of zero. For the default run: 1 × (−3.46) + 3 × (+1.15) = −3.46 + 3.45 ≈ 0 kg·m/s, matching the p_tot readout within numerical precision. For the equal-mass case (m₁ = m₂ = 2 kg), the readout confirms p_tot = 0.00 kg·m/s with both carts moving at equal and opposite speeds.

Key Variables

The table below lists the physical quantities that appear in Newton's third and second law equations as they are used in the cart simulator. Symbols match the readout labels in the simulation panel.

Real World Examples

How does Newton's third law govern a rocket engine in space?

A rocket engine works entirely by Newton's third law. Hot exhaust gas is expelled backward at high speed; the rocket body experiences the equal and opposite force forward. There is no air for the engine to push against — the reaction force acts directly on the rocket through the pressure difference across the combustion chamber.

The mass ratio here is extreme: exhaust molecules are vastly lighter than the rocket body, so the exhaust reaches enormous speeds while the rocket accelerates more slowly, exactly as the cart simulator shows when m₁ = 1 kg and m₂ = 5 kg — the lighter cart (analogous to exhaust) leaves at high speed while the heavy cart moves slowly in the opposite direction. With m₁ = 1 kg and m₂ = 5 kg in the simulator, the v₁ readout shows approximately −3.65 m/s while v₂ shows approximately +0.73 m/s — a 5:1 speed ratio matching the inverse mass ratio, p_tot remaining at 0.00 kg·m/s throughout.

Rocket engineers express this through the Tsiolkovsky rocket equation, which integrates the continuous version of the same action-reaction relationship. The cart simulation captures the discrete impulse version of the same physics: the spring delivers one fixed impulse, and the velocity ratio is the inverse of the mass ratio. Real rockets apply that ratio continuously as propellant burns off and the vehicle mass decreases, but the underlying third-law pairing is identical.

Why does an astronaut pushing off a spacecraft drift away slowly while the spacecraft barely moves?

When an astronaut pushes off a spacecraft, the contact force during the push is the same on both objects — Newton's third law guarantees that. The spacecraft is thousands of times more massive than the astronaut, so its acceleration under that shared force is thousands of times smaller. The astronaut drifts away at a noticeable speed; the spacecraft recoils by a distance too small to measure without instruments.

This is the extreme mass-ratio limit of the cart system. Setting m₁ = 1 kg and m₂ = 5 kg in the simulator illustrates the principle at a more modest ratio: the v₁ readout reaches approximately −3.65 m/s while v₂ reaches approximately +0.73 m/s, and the p_tot readout stays at 0.00 kg·m/s. Scaling the mass ratio from 5:1 to the realistic thousands-to-one simply pushes the spacecraft speed closer and closer to zero while the impulse stays equal.

The practical consequence for spacewalking astronauts is that any push — against a handrail, a fellow crew member, or a piece of equipment — imparts momentum to both parties. Mission planners account for this when scheduling extravehicular activities: unplanned pushes can send an untethered astronaut drifting away from the station at a fraction of a metre per second, a slow speed by everyday standards but one that compounds over minutes into a dangerous separation distance.

How do engineers use the action-reaction principle to design ejection seats?

An ejection seat must accelerate a pilot out of a cockpit in well under a second — typically reaching 18 to 20 g of acceleration upward. The propellant charge fires and pushes the seat and pilot upward; the equal reaction force pushes the aircraft downward, though the aircraft is far too massive to move noticeably. Engineers treat the seat-plus-pilot as one mass and compute the required impulse using J = F · t, then choose propellant charge, burn time, and seat mass to hit the target exit speed without exceeding the pilot's spinal tolerance.

The cart simulator uses the same impulse framework: with the default m₁ = 1 kg and m₂ = 3 kg, the spring delivers a shared impulse of roughly 3.46 N·s, producing v₁ ≈ −3.46 m/s and v₂ ≈ +1.15 m/s. The p_tot readout confirms the impulses cancel to zero throughout. Ejection-seat designers solve the same equation in reverse — specifying the required exit velocity and back-calculating the charge parameters.

Modern zero-zero ejection seats (capable of saving a pilot at zero altitude and zero forward speed) must deliver the impulse so rapidly that the seat clears the aircraft before the parachute deploys. The mass of the seat and pilot combined is the m₁ in the action-reaction equation; the aircraft is the m₂. Because the aircraft mass is orders of magnitude larger, the aircraft velocity change is negligible — but the impulse on the seat is real, large, and precisely engineered.

Further Reading

The following articles on this site extend the concepts introduced here, moving from the isolated two-cart push to more complex force and motion scenarios that build on the same Newton's-law foundation.

• Newton's first law — the puck — inertia and the tendency of objects to maintain constant velocity in the absence of net force, the foundation that makes the cart push meaningful.
• Inelastic collisions — what happens when two objects stick together after contact; momentum is still conserved but kinetic energy is not, in contrast to the spring-release case.
• Friction block — introduces a horizontal resistance force that modifies the net force in Newton's second law, showing how real surfaces deviate from the frictionless track assumed here.