Newton's third law

Write every force with both objects in its name: F(blocks on sprinter), F(Earth on sprinter). The first name is the agent and the second is the object the force acts on, and only the second decides whose free-body diagram the arrow joins. The labels settle arguments that intuition loses. A 68 kg sprinter leaves the blocks accelerating horizontally at 4.0 m s⁻², so something outside her must push her forward with 68 × 4.0 = 272 N. Her muscles cannot be that agent: they are inside her, and forces inside a body come in pairs that cancel. The agent is the starting block, F(blocks on sprinter) = 272 N forward. What her legs actually do is push the block, F(sprinter on blocks) = 272 N backward, and that arrow belongs on the block's diagram, not hers. If you cannot name the agent of a force, you are about to draw a force that does not exist.

To find a partner, swap the two names and reverse the direction; nothing else changes. Four checks expose every false pair: the partner acts on the other object, has the same magnitude, points the opposite way, and comes from the same interaction, so it is the same type of force. Because the law is a vector equation it holds component by component. A 20 kg crate sliding across a floor with a kinetic friction coefficient of 0.30 receives one contact force from the floor, made of a 196 N normal part pushing up and a 58.9 N friction part pointing backward: 205 N in all, tilted 16.7° back from the vertical. Its partner is the crate's contact force on the floor, 196 N down and 58.9 N forward. The floor is dragged forward by the crate, which is why a loose rug creeps the same way as a box you shove across it.

Forces cancel only when they act on the same object. A third-law pair never does, so a pair can never explain why something stays still, and it never stops anything from accelerating. A 65 kg shopper pushes a 40 kg trolley with 36 N, and the trolley pushes back on her with 36 N at every instant, yet both speed up together. Each object obeys its own diagram. The trolley feels 36 N forward and 12 N of rolling resistance, so it accelerates at (36 − 12)/40 = 0.60 m s⁻². The shopper feels the trolley's 36 N backward and the floor's friction on her shoes forward; to keep pace at 0.60 m s⁻² the floor must push her with 36 + 65 × 0.60 = 75 N. The partner of that 75 N is her shoes' 75 N backward push on the floor, which is what walking is.

A pair shares a magnitude, not an effect. Each object responds with its own acceleration, a = F/m, so the lighter object takes almost all of the motion. A 0.50 kg ball dropped from 1.2 m pulls Earth with 4.9 N, exactly as hard as Earth pulls it. The ball accelerates at 9.81 m s⁻²; Earth, of mass 5.97 × 10²⁴ kg, accelerates toward the ball at 4.9 N ÷ (5.97 × 10²⁴ kg) = 8.2 × 10⁻²⁵ m s⁻². Both start from rest and the pull acts on both for the same 0.49 s, so their displacements stand in the ratio of their accelerations: Earth rises m h/M = 0.50 × 1.2 ÷ (5.97 × 10²⁴) = 1.0 × 10⁻²⁵ m, more than ten billion times less than the width of a proton. That is why the partner of a weight is so easy to forget. It is real and exactly equal, and its effect on Earth cannot be measured.

There is no delay inside a contact pair. Neither force comes first, and the words action and reaction only record which half you happened to mention first. Because the two forces match at each instant of a contact, they also match when integrated over it. A 0.145 kg ball arriving at 38 m s⁻¹ leaves the bat at 45 m s⁻¹ in the opposite direction, so the bat gave it an impulse of 0.145 × (38 + 45) = 12.0 N s. Over 0.70 ms of contact that is an average force of 12.0 ÷ 0.00070 = 1.7 × 10⁴ N on the ball, and the ball delivers the same 12.0 N s to the bat, pointing the other way. The bat's momentum does not change by exactly that amount, because the batter's hands push on it too. The third law pairs the impulses; momentum is conserved only for a system that nothing outside pushes.

Two situations break the law itself, and a third only looks as though it does. Fictitious forces have no partner: the outward push you seem to feel in a turning car has no agent, so no object receives its reaction, and that missing partner is the sign that the car is not an inertial frame. Electromagnetic forces between moving charges act through a field. When one charge moves straight along the line joining it to a second charge that moves across that line, the first feels a magnetic force from the second while exerting none back. Momentum is still conserved, but only once the field's own momentum is counted. The false alarm is a connector with mass, which does not hand one force along unchanged. Drag a 15 kg sled across frictionless ice with a 0.60 kg rope at 2.0 m s⁻² and the rope pulls the sled with 15 × 2.0 = 30.0 N, while the hand pulls the rope with 15.6 × 2.0 = 31.2 N; the 1.2 N difference accelerates the rope. No pair is unequal there: the true pairs are hand with rope and rope with sled, because the hand and the sled never touch.