Newton's three laws of motion: push a cart and see them work

How to use this simulation

1. Pick "1st law: inertia" and press play: the push lasts 2 s, then the force is gone, yet on ice the cart glides on at the same speed.
2. Watch the passenger box when the cart slams into the wall: it slides forward. Turn the force right up and it slides backward at the start instead.
3. Switch to "2nd law: F = ma": the slope of the v–t graph is the acceleration. Double the mass and the new line climbs at half the slope of the grey previous run.
4. Turn friction on and lower the force: while F is below μmg the cart does not move at all, and the readings show friction exactly equal to the push.
5. Choose "3rd law" and give the two carts different masses: the pushes are equal and opposite, the lighter cart moves faster, and total momentum stays at zero.

Why you lurch forward when the bus brakes

You are standing on a crowded bus and the driver slams on the brakes. Nobody pushes you, yet you stumble forward. When the bus pulls away suddenly, you tip backward instead. Where do those invisible shoves come from?

A football lying on the pitch never rolls into the goal by itself; somebody has to kick it, and the harder the kick the faster it goes. But kick a brick with the same effort and it barely moves, and your toe hurts. Why does the same effort do such different things?

Jump from a small boat to the shore and the boat shoots backward. Swim, and you move forward only because your hands push water back. All of these everyday puzzles are answered by three short rules that Isaac Newton published in 1687 in his book the Principia. The simulation above lets you test all three with a cart on a track.

Starting from zero: what a force is

A force is a push or a pull. It is what changes an object's state of rest or of motion: it can start something moving, stop it, speed it up, slow it down or turn it. Force has a size and a direction, so it is a vector, and its SI unit is the newton (N).

For about two thousand years people followed Aristotle in believing that a moving object needs a constant push to keep moving, because that is what we see: switch off a car engine and the car soon stops. Galileo was the first to realise that things stop because of friction, not because motion "runs out". The smoother the surface, the further a ball rolls; on a perfectly frictionless surface it would never stop.

Newton turned that insight into three laws. The first says what happens when there is no net force. The second says how much motion changes when there is one. The third says that forces always come in pairs. Between them they explain nearly every motion you will meet at school, from a bicycle to a rocket.

Key words for Newton's laws

Definition questions come straight from this table, so read it once carefully.

Newton's first law of motion: the law of inertia

Statement: an object remains at rest, or keeps moving in a straight line at constant speed, unless it is acted on by an unbalanced (net) external force.

The first half feels obvious: a book on a table stays put until someone moves it. The second half is the surprise: a moving object does not slow down on its own. Whenever something does slow down, a force is doing it, usually friction or air resistance.

The cart on ice shows exactly this. A 5 kg cart is pushed with 10 N for 2 s. Then the force drops to zero, but the cart carries on at 4 m/s until it reaches the wall. Its speed neither grows nor shrinks, because the net force is zero, so that stretch of the v–t graph is perfectly flat.

The first law also gives us the definition of force: whatever changes an object's state of rest or motion. The property of an object that resists such change is called inertia, which is why the first law is also called the law of inertia.

Balanced and unbalanced forces

When several forces act on one object, their combined effect is the resultant or net force. If the net force is zero the forces are balanced and the motion does not change: at rest stays at rest, moving stays moving at constant velocity. If the net force is not zero the forces are unbalanced and the object accelerates.

A book on a table feels its weight pulling down and the table's normal force pushing up. They are equal and opposite, so they balance and the book stays still. In the simulation the 5 kg cart has weight W = mg = 49 N, and the green normal-force arrow N is also 49 N; the two arrows are drawn the same length.

Turn friction on (μ = 0.1) and apply 4 N. The cart does not budge, because static friction grows to exactly 4 N in the opposite direction: push and friction balance. Raise the push above 4.9 N and the balance breaks, so the cart starts to move.

Notice that moving at constant velocity also means balanced forces. After the push the cart on ice is moving, yet the net force on it is zero. "It is moving, so something must be pushing it forward" is Aristotle's mistake, and it still costs marks.

Momentum: the quantity of motion

Momentum is mass multiplied by velocity: p = mv. It points the same way as the velocity, so it is a vector. Its unit is kg·m/s, which is the same as N·s.

Momentum tells you how hard something is to stop. A bicycle and a lorry moving at the same speed are very different to stop, because the lorry has far more mass. A bullet has very little mass, but its speed is so great that its momentum is dangerous.

At the end of the simulation's push the 5 kg cart moves at 4 m/s, so its momentum is p = 5 × 4 = 20 kg·m/s, the value shown in the momentum reading.

Newton's second law of motion and F = ma

Statement: the rate of change of momentum of an object is directly proportional to the net force applied, and takes place in the direction of that force.

In plain words: the bigger the force, the faster the velocity changes; the bigger the mass, the more slowly the same force changes it. From this law comes the most famous equation in physics, F = ma.

Impulse: why a fielder pulls the hands back

Some forces are very large but act for a very short time: a bat hitting a ball, a hammer on a nail, a kick. They are called impulsive forces. Because the force changes so fast it is hard to measure, so we measure force × time instead, called the impulse: J = F × t.

Rearrange the second law F = (mv − mu)/t and you get F × t = mv − mu. Impulse equals the change in momentum. That gives a powerful trick: if the change in momentum is fixed, stretching the time makes the force smaller.

Why does a cricket fielder pull the hands back while catching? A 0.16 kg ball arriving at 30 m/s (about 108 km/h) has momentum 4.8 kg·m/s. Stopped in 0.05 s by stiff hands, it pushes on them with 96 N. Pull the hands back so it stops in 0.4 s and the force is only 12 N, 8 times smaller. It stings less, and the ball is less likely to bounce out.

The same idea is behind bending your knees when you land from a jump, the thick mat behind a high jump, foam packaging around glassware, crumple zones in cars and airbags. Every one of them stretches the stopping time to cut the force.

Newton's third law of motion: action and reaction

Statement: to every action there is an equal and opposite reaction. More exactly: when object A exerts a force on object B, B exerts a force of the same size in the opposite direction on A, at the same instant.

Both the action and the reaction are forces, and which one you call the "action" is only a label. They are born together and vanish together. They are also always the same kind of force: a contact push is answered by a contact push, a gravitational pull by a gravitational pull.

Punch a wall and your hand hurts because the wall pushes back just as hard. When you walk, your foot pushes the ground backward and the ground pushes you forward. An oar pushes water back and the water pushes the boat forward. A bird pushes air down with its wings and the air pushes the bird up.

Conservation of momentum: guns and rockets

When two objects push on each other, the third law makes the forces equal and opposite, and they act for the same time. So whatever momentum one gains, the other loses. With no external force, the total momentum of the objects never changes. That is the law of conservation of momentum.

In the simulation both carts start at rest, so the total momentum is zero. After the push cart 1 has −5 kg·m/s and cart 2 has +5 kg·m/s, which again add to zero. Watch the total-momentum reading: whatever masses you choose, it stays at zero while friction is off. Switch friction on and it drifts, because friction is an external force.

Gun recoil is the same story. Before firing, gun and bullet are at rest. A 0.02 kg bullet leaving at 400 m/s carries 8 kg·m/s forward, so the 4 kg gun must move back at 2 m/s to keep the total at zero. That is why a rifle is held firmly against the shoulder.

A rocket throws hot gas backward at enormous speed; the gas carries momentum backward and the rocket gains the same momentum forward. It needs no air to push against, which is why rockets work in space. Blow up a balloon and let it go, and it zooms off for the same reason.

Friction and free-body diagrams

Friction is the force that appears between two surfaces that slide, or try to slide, over each other. It always opposes the relative motion. Static friction acts before anything moves and can grow up to a limit; kinetic (sliding) friction acts once it slides; rolling friction is much smaller, which is why wheels were such a good idea; and fluid friction (drag) acts in water and air.

In the simulation friction is f = μmg. For the 5 kg cart with μ = 0.1, f = 0.1 × 5 × 9.8 = 4.9 N. To keep things simple the same μ is used for static and kinetic friction; in reality the static limit is usually a little higher.

To see which forces act on an object, draw it as a box or dot and draw every force on it as an arrow starting from it. That picture is a free-body diagram, and it is the first step in almost every mechanics problem. The arrows on the cart are exactly such a diagram, all drawn to one scale: double a force and its arrow doubles.

Try these in the simulation

Grab a notebook. Before each run, write down what you expect; then run it and check.

• In first-law mode with friction off, press play. After 2 s the label reads "No push, gliding on its own", yet the velocity stays at 4 m/s.
• Now switch friction on. When the push ends the cart is at 2.04 m/s; friction then removes 0.98 m/s every second, and it stops about 2.08 s later after 2.12 m. Friction stopped it, not a lack of force.
• Set the force to 50 N and the mass to 1 kg: the sudden start makes the passenger box slide backward (inertia of rest), and the crash into the wall makes it slide forward (inertia of motion).
• In second-law mode run 5 kg with 10 N, then change the mass to 10 kg and run again. Is the new slope half the old one?
• With friction on, set the force to 4 N: the cart stays put and "Static friction is holding it back" appears. At what force does it start moving? Answer: just above 4.9 N.
• In third-law mode give both carts the same mass: they move off at equal speeds in opposite directions. Make one four times heavier: its speed is a quarter. Does the total momentum ever show anything other than zero?

Solved problems

Each solution lists what is given, then the formula, then the numbers. Set out answers the same way in an exam and you will not lose method marks.

At a glance: one force, different masses

10 N applied for 2 s on a frictionless track. Look at the last column: whatever the mass, the same force for the same time gives the same momentum, because the change in momentum equals the impulse.

Common mistakes

Avoid these and Newton's-laws questions become easy marks.

• Thinking a moving object always has a forward force on it. At constant velocity the net force is zero.
• Cancelling an action against its reaction. They act on different objects, so they never cancel.
• Calling weight and the normal force an action–reaction pair. Both act on the same object, so they are balanced forces.
• Putting the applied force into F = ma when friction is present. Use the net force, F − f.
• Treating inertia as a force. Inertia is a property of matter, not a push or a pull.
• Mixing up mass and weight. Mass is in kg and is the same everywhere; weight is mg, in newtons, and changes from place to place.
• Forgetting that momentum has a direction. Take velocities in the opposite direction as negative, or conservation sums will not balance.

Newton's laws in real life

Once you know the three laws, you will spot them everywhere.

• Seat belts and airbags hold a passenger whose inertia carries them forward, and stretch the stopping time to cut the force.
• Sport: pulling the hands back to catch, padded goalkeeper gloves and follow-through in a golf swing are all impulse calculations.
• Rockets and jet engines throw gas backward to move forward: the third law and conservation of momentum.
• Walking, swimming and rowing: push the ground or water backward and the reaction pushes you forward.
• Heavy lorries need longer stopping distances because more mass means more inertia, which is why drivers keep a bigger gap behind them.
• Shaking a branch to drop fruit, or flicking water off wet hands, uses inertia of rest and of motion.

Exam corner

Newton's laws sit at the heart of every school mechanics course, from Grade 9 forces and motion to Grade 11 dynamics, and they underpin later topics such as circular motion, gravitation and collisions. Expect three kinds of question: state-and-explain (write each law, define inertia, momentum and the newton), explain-an-observation (why a passenger lurches, why a fielder pulls the hands back, why a gun recoils) and calculations (F = ma with friction, impulse, conservation of momentum).

Exam technique that earns marks: always draw a free-body diagram before writing any equation; use the net force in F = ma; choose a positive direction and stick to it; and in a third-law question, name both objects ("the ground pushes the foot, the foot pushes the ground"). In multiple-choice questions, the classic traps are the horse-and-cart paradox, "the heavier object exerts the bigger force in a collision" (it does not; the forces are equal) and confusing momentum with kinetic energy.

Revision: the last-minute summary

Read this list once the night before the exam.

• First law: with no net external force, an object at rest stays at rest and a moving object keeps a constant velocity; it is the law of inertia.
• Inertia comes as inertia of rest and inertia of motion (some books add inertia of direction); mass measures it.
• Balanced forces have zero resultant and change nothing; an unbalanced force causes acceleration.
• Momentum p = mv, in kg·m/s; second law: rate of change of momentum ∝ net force, giving F = ma.
• 1 N = 1 kg·m/s²; impulse J = Ft = change in momentum; a longer stopping time means a smaller force.
• Third law: every action has an equal and opposite reaction, on a different object, so the pair never cancels.
• With no external force, total momentum is conserved: gun recoil, rockets, carts pushing apart.
• Friction f = μmg opposes motion; put the net force into F = ma.

Frequently asked questions