Three identical balls are at the same height. One is going up. One is at the top, at rest for an instant. One is coming down. Ignore air resistance.
Are the forces on the three balls the same?
They are. Each ball has the same force on it, its weight, pointing down. The motions differ and the forces do not.
In Viennot’s samples, between 30 and 55 percent of students got this wrong. The pattern held from secondary school to third-year university, and across France, Britain and Belgium. The most common reasons were that the forces are different because the motions are, or that the force is zero because the velocity is zero.
These are not careless mistakes. The same wrong answers come up in group after group, for the same reasons.
Why the second law is hard
Newton’s second law relates quantities at the same instant. The net force on a body now equals its mass times its acceleration now. Nothing that happened earlier enters the equation.
This is routine for physicists. For students it is not obvious, because their natural way of explaining motion is a story. First comes a cause, such as a throw or a push. Then, later, comes the effect, and the object moves. Cause first and effect later is how ordinary explanation works. The second law does not work that way.
Laurence Viennot put it like this. A law relating simultaneous quantities is a law and not a story. Students want a story. When the story does not match the physics, they keep the story and change the physics.
What is happening in their heads
Take a ball thrown upward. In the student’s version, the thrower gave the ball force. The ball carries this force upward. The force runs out, and the ball falls.
As physics, this is wrong. Force is not something you give to an object. Objects do not carry force, and force does not get used up. At the top of the flight gravity is still acting, so the force is not zero.
The reasoning holds together in the student’s mind, though. Listen to the language students use, year after year:
- “The force of the thrower’s movement is still at work at the top.”
- “The mass has a force.”
- “A supply of upward force.”
- “The force the fellow gave it.”
Three things are happening at once.
First, force is treated as something the object has. In physics, a force describes an interaction between two objects, the force of A on B. The student says “the force of the mass”, as if force were a property of the object. Once force is something an object owns, it can be stored, carried and used up.
Second, quantities from the past are pulled into the present. The thrower let go of the ball seconds ago, yet “the force of the thrower” is called on at the top. The student takes a past event and installs it in a present explanation.
Third, the lines between different quantities fade. Students use “force”, “energy”, “velocity”, “inertia” and “impetus” almost interchangeably. Some draw diagrams that add a force arrow to a velocity arrow, which mixes two different kinds of quantity. Inside the student’s picture these are all versions of one idea, the object’s capacity to keep doing what it is doing.
Viennot calls this “dynamism”. Students do not hold a tidy rival theory by that name. They reach for the same vague causal resource whenever motion needs explaining.
The error is selective
If the confusion between force and velocity were plain ignorance, it would show up evenly across problems. It does not.
Viennot found that the error rises under two conditions. The motion must be vivid and easy to picture. And the motion must look incompatible with the known forces.
The pendulum shows this. It is one device with the same two forces, weight and the tension in the string. Two situations show the contrast best.
Take a mass at the top of a full circular loop. Weight and tension both point downward, yet the object carries on round the loop. This has the highest error rate. Two forces pull down, nothing visible pushes up, and the object does not drop straight down.
Students invent what they need, such as a centrifugal force pointing outward, or a tension that has reversed direction. They build forces on the spot to make sense of the situation on their own terms.
Now take a mass at the highest point of an ordinary swing. It is at rest for an instant, with a net force on it that is not zero. This has the lowest error rate. Zero velocity with a non-zero force should be just as troubling. It is not, because everyday reasoning accepts a short delay between cause and effect. The force is there, and it will produce motion soon.
What everyday reasoning cannot accept is motion that keeps going with no visible cause.
This selectivity matters. The errors are not failures of memory. They come from a way of reasoning that brings in invented forces, stored dynamism and imported past causes. It does so wherever the known forces seem unable to explain the motion.
The same logic in waves
If this pattern were confined to thrown balls and pendulums, you could treat it as a narrow mechanics problem. It is not.
Ask students whether the speed of a bump on an idealised rope depends on how hard you flick the rope. Many say yes. A harder flick puts more force in the bump, so it moves faster.
Ask whether a bigger bump overtakes a smaller one. Many say yes, because it has more force. Ask what happens as the bump flattens. They say it has less and less force, so it moves more and more slowly, because of friction.
The student has invented an object. A bump on a rope is a disturbance moving along the rope, and not a material body. Everyday reasoning needs an object to carry force, so the bump becomes one. It is loaded with stored dynamism from the flick, and then it wears out. The story is the same as the thrown ball.
Sound does the same thing. Students predict that a louder shout makes sound travel faster, because the sound has more force. Some think sound can travel through a vacuum, since there is nothing in the way to stop it.
For teaching, stating the rule is not enough. For this idealised rope, pulse speed is set by tension and by mass per unit length. Students can recite that and still hold the underlying picture.
The rule has to be set against the picture it replaces. The speed of the bump does not depend on the hand motion, in this idealised case. That surprises students, and the surprise is the point. Then explain why the rule holds.
Why the third law is hard
Once force belongs to objects, and explanation is organised around stories of stored cause, Newton’s third law becomes hard to accept for a structural reason.
A stronger student and a weaker student push against each other on low-friction carts. Common sense says the stronger student exerts the larger force. Newton says the force of A on B and the force of B on A are equal in size and opposite in direction. If their accelerations differ, that is because their masses differ or other external forces act. It is never because the pair is unequal.
If each person “has a force” of their own, in proportion to their strength, then equal forces are absurd. The stronger one must push harder, or the word stronger means nothing.
This is more than a gut feeling getting in the way. If force belongs to objects, the third law makes no sense at all. Only when force is an interaction do equal and opposite paired forces make sense. And seeing force as an interaction means giving up the causal picture that makes mechanics feel natural.
Two things get mixed together, and separating them is half the work:
- The forces acting on one object, which add up to decide its acceleration.
- The pair of forces in one interaction, which act on two different objects and are always equal and opposite.
These are different operations. Students fold them into one, and standard force diagrams often let them.
Viennot’s fix is the fragmented diagram. Draw the interacting objects apart, even when they touch. For each interaction, draw one arrow on each object, in the same colour and the same length, pointing opposite ways. Then, separately, draw the force balance on each object by adding up all the forces on it. The procedure makes the confusion much harder to keep up.
Situations with acceleration teach this better than situations in equilibrium. In a book-on-table diagram, the weight of the book and the normal force on the book are equal in size. Students may then call them an action and reaction pair, even though both act on the same object. Cases with acceleration remove that ambiguity.
For this one distinction, forces at a distance are easier than contact forces. With no contact, you have to decide which object each force acts on. Contact creates an ambiguity that distance removes.
The spring problem
A problem in Viennot’s appendices puts the whole argument in one place.
A mass sits on a spring. It is not attached, only resting. You push the mass down, compressing the spring, and let go.
How far down must you push for the mass to fly off the spring?
Students say the spring force at maximum compression must exceed the weight.
At the instant the mass lifts off, the spring is at its natural length, so the spring force is zero. The only force acting is gravity, pointing down. The mass rises with no upward force on it.
So the compression must be large enough that the mass still has upward kinetic energy when the spring reaches its natural length. A spring force that merely exceeds the weight at maximum compression is not enough.
Students find this hard to accept, so they reach into the past. They compare the spring force from a moment ago with the weight from right now, as if Newton’s law allowed a balance across time. When the comparison fails, some conclude that “the law of action and reaction no longer applies”.
The student’s reasoning demands a cause that is present now whenever the visible motion seems unexplained. Physics says the explanation lies in the object’s state at that instant, and not in a cause carried over from the past.
When the student’s picture breaks on that point, they import a past cause anyway. That breaks the time structure of the law, and when the analysis fails they throw out the third law. The whole argument is in that one problem.
What to do about it
The aim is to see the system behind the errors, and not to collect them.
Students arrive with a consistent way of making motion make sense. It runs across dynamics, wave motion, support forces and the analysis of interactions, and stating the right formulas does not take it apart. The system has its own logic.
Objects have dynamism, and dynamism explains motion. It comes from past causes and it fades over time. This logic works for much of everyday life, where people get tired, batteries run down, and cars slow down when you lift off the accelerator.
Newtonian physics works differently. Force is an interaction and not a possession. Laws relate quantities at the same instant, and not a cause to a later effect. The past matters only through the state it has brought the system to now.
Viennot’s teaching principle is easy to state and hard to carry out. Whenever a correct result contradicts what natural reasoning expects, say so out loud. Do not state the rule and assume the student sees why it is surprising. Say what the surprise is.
- “The speed of the bump does not depend on the initial hand motion, and that is strange.”
- “On the way up, after release, the only force on the ball is gravity, even though the ball is still moving upward.”
- “The force the hammer exerts on the nail equals the force the nail exerts on the hammer, even though the nail goes in.”
This asks the teacher to know, at every point, what the student’s alternative reasoning predicts. A teacher who does not understand the student’s reasoning cannot make the conflict visible, and a conflict that is not visible will not be resolved.
This article draws on Laurence Viennot, Reasoning in Physics: The Part of Common Sense (Kluwer, 2001), especially Chapter 4, “The essential: laws for quantities at time t”, and on the earlier studies of spontaneous reasoning in dynamics and wave propagation by Viennot and Laurence Maurines.