Newton's first law

Before this

This leads to

Newton's second law
Free-body diagrams

What keeps a moving car moving?

A car drives along a straight, level highway at a steady 90 km/h. Its speed does not change, and its direction does not change. Here is the question: what does the engine's force have to do to keep the car moving like this?

Most people answer at once: the engine must push hard, because the car is moving fast. If the engine stopped pushing, the car would slow down. So the push must be what keeps it moving.

That answer feels like common sense. It is also wrong, and it was believed by careful thinkers for nearly two thousand years. The correct idea is simple to state but hard to believe at first: an object does not need a resultant force to keep moving. It needs a resultant force only to change how it moves. This lecture is about that one idea. It is the first of Newton's three laws, and every later lecture on forces is built on it.

Aristotle's idea, and why it seems true

The Greek thinker Aristotle taught that the natural state of an object on Earth is rest. To keep something moving, he said, something must keep pushing it. Stop pushing, and it stops.

Everyday life seems to agree. Push a heavy box across a floor and let go: it slides a little way and stops. Motion always seems to die away unless something keeps it going.

But look more closely at the box. When you let go, is nothing acting on it? Something is. The floor rubs against the bottom of the box, and that rubbing is a force: friction. Friction acts backwards, against the motion. The box stops because a force stops it, not because the push has run out.

In the early 1600s, Galileo Galilei argued this from experiments with balls rolling on smooth ramps. The smoother he made the surfaces, the further a ball rolled before stopping. He reasoned that on a perfectly smooth, level surface, the ball would roll forever. Isaac Newton made this the first law of his laws of motion, published in 1687.

You can test the same idea below. A puck slides across different surfaces after one short push.

Simulation — one push, then let go

The puck (mass 0.5 kg) leaves your hand at 4 m/s. Choose a surface and press Play, or Step. Then make the surface smoother and run it again. Finally, tick "keep pushing" and see what a push equal to the friction does.

Surface: wood

Keep pushing with a force equal to the friction

The puck's position every second after it leaves the hand at 4 m/s, on four surfaces:

On wood, friction (0.49 N) stops the puck after about 4.1 s and 8.2 m. On a smooth floor it slides about 16.3 m, and on ice about 40.8 m. With no friction, the dots stay equally spaced: the puck moves 4 m every second and never slows. A push exactly equal to the friction gives the same result on any surface.

Did you see the pattern? On rough carpet the puck stops after about 4.1 m; on ice it slides about 40.8 m. With no friction, it keeps its 4 m/s for as long as the track lasts. And a push equal to the friction gave a steady 4 m/s on every surface. The push did not speed the puck up. It only cancelled the friction.

Newton's first law of motion

An object stays at rest, or keeps moving at constant velocity in a straight line, unless a resultant force acts on it.

Every phrase in the law matters, so take them one at a time.

"Stays at rest, or keeps moving at constant velocity." These look like two different cases, but the law treats them as one. Rest is simply constant velocity with the value zero. Without a resultant force, the velocity does not change, whatever it is.

"In a straight line." Velocity has a size and a direction. Constant velocity means constant speed and constant direction. A car going round a bend at a steady 30 km/h has a changing direction. Its velocity is changing, so by the first law a resultant force must be acting on it. A later lecture on circular motion studies that force.

"Resultant force." This is the most important phrase. The resultant force is the single force that has the same effect as all the forces on the object together. (Some books call it the net force.) The law does not say "unless a force acts". It says "unless a resultant force acts". An object can have many forces on it. If they cancel, the resultant is zero, and the object behaves exactly as if no force acted at all.

Read the law the other way round, and it says something just as useful. If the velocity of an object is changing, there must be a resultant force on it. Speeding up, slowing down and changing direction all need a resultant force. Moving steadily in a straight line does not. That is it. That is the whole idea. The next lecture, on Newton's second law, works out how large the change in velocity is for a given resultant force.

Balanced forces

Now return to the car on the highway. Four forces act on it.

The forces on an invented 1200 kg car at a steady 90 km/h. Up and down, the road's push (the normal force) balances the weight. Forwards and backwards, the driving force balances air resistance and friction together. The arrows are not to scale: 600 N is far smaller than 11 760 N.

Weight, the pull of the Earth, acts downwards. For a 1200 kg car it is $m g = 1200 \times 9.8 = 11760 N$ .
The normal force from the road acts upwards. The car is not sinking into the road or rising from it, so this must also be 11 760 N.
The driving force, from the road pushing on the turning tyres, acts forwards.
The resistive forces, mainly air resistance plus some friction, act backwards.

The car's velocity is constant, so the resultant force must be zero. That means the driving force must exactly equal the resistive forces. If these come to 600 N, the driving force is also 600 N.

Stop and think

So does the engine provide zero force when the car moves at a steady 90 km/h?

Show the answer

No. The engine still drives the wheels, and the driving force is 600 N. What is zero is the resultant force. The driving force is needed only because air resistance and friction push backwards. Without the driving force, the resultant would be 600 N backwards, and the car would slow down. In empty space, with no air and no road, the car would need no driving force at all to keep its speed. Always ask "zero force, or zero resultant force?" They are very different statements.

Space gives the clearest example. The space probe Voyager 1 was launched in 1977. Its rockets stopped pushing it long ago, yet it is still moving away from the Sun at about 17 km/s, according to NASA. Almost nothing acts on it. The Sun's gravity, very weak at that distance, slows it only very slightly.

Inertia: why some objects are harder to change

The first law says that objects resist changes to their motion. This tendency has a name: inertia. An object at rest tends to stay at rest. A moving object tends to keep moving with the same velocity.

Some objects have more inertia than others. Push a table-tennis ball and a bowling ball with the same gentle push. The table-tennis ball shoots away, and the bowling ball barely moves. Now try to stop each one when it rolls towards you. The bowling ball is much harder to stop. It has more inertia.

The quantity that measures inertia is mass. More mass means more inertia: a bigger resultant force is needed to change the motion by the same amount. This is different from weight. On the Moon, a bowling ball weighs about one-sixth as much, but it has the same mass. It would be just as hard to start rolling sideways and just as hard to stop.

Falling at a steady speed

A skydiver jumps from a plane. At first, only one force matters: her weight, pulling her down. The resultant force is downwards, so she speeds up. But as her speed grows, air resistance grows too, because she hits more air every second, and hits it harder.

A velocity-time graph for a 75 kg skydiver (a simplified, invented model). A: weight is larger than air resistance, so she speeds up. B: the two forces are equal, so her velocity is constant at about 54 m/s. C: the parachute opens, air resistance becomes much larger than her weight, and she slows down. D: the forces balance again at about 5 m/s.

Eventually air resistance grows until it equals her weight. The resultant force is now zero. By the first law, her velocity stops changing. She falls at a constant speed called her terminal velocity, about 54 m/s in this model.

Notice what has not happened. She has not become weightless. Her weight is exactly what it was when she jumped. It is now balanced by an equal upward force.

Use the first law on the next part too. When the parachute opens, air resistance suddenly becomes far larger than her weight. The resultant force now points upwards. But she is still moving downwards. An upward resultant force on a falling object does not make it go up. It makes it slow down. As she slows, air resistance falls, until it balances her weight again at a new, much lower terminal velocity.

Check your understanding of resultant force with the cards below.

Sorter — is the resultant force zero?

For each situation, decide whether the resultant force on the object is zero. Ask one question: is the velocity (speed or direction) changing?

Each situation, and whether the resultant force on the object is zero:

Situation	Resultant force
A car at a steady 90 km/h on a straight, level road	zero
A car going round a roundabout at a steady 30 km/h	not zero (direction changing)
A skydiver falling at terminal velocity	zero
A ball at the top of its flight, thrown straight up	not zero (its weight)
A lift moving up at a steady speed	zero
A book resting on a table	zero
A bus braking at a red light	not zero (slowing)
A space probe far from any star, engines off	zero (very nearly)
A cyclist speeding up away from a junction	not zero (speeding up)

Worked examples

Worked Example 1 — A book at rest

A book of mass 1.5 kg rests on a table. Identify the forces on the book and explain why it does not move.

Weight acts downwards: $W = m g = 1.5 \times 9.8 = 14.7 N$ . The table pushes up on the book with a normal force $N$ .

The book stays at rest, so its velocity is not changing. By the first law, the resultant force is zero. So the normal force must be 14.7 N upwards.

The book also pushes down on the table. That force acts on the table, not on the book, so it does not belong in this list. A later lecture, on Newton's third law, explains such pairs of forces.

Check: 14.7 N up and 14.7 N down give a resultant of zero, which matches a book that stays at rest.

Worked Example 2 — The car speeds up, then settles

The invented car moves at a steady 25 m/s (90 km/h) with a driving force of 600 N. The driver presses the accelerator, and the driving force rises to 900 N. Describe what happens.

At first, the resistive forces are still 600 N. The resultant force is $900 - 600 = 300 N$ forwards. A resultant force means a changing velocity, so the car speeds up.

As the car goes faster, air resistance grows. The resultant force gets smaller and smaller. When the resistive forces reach 900 N, the resultant is zero again, and the car moves at a new, higher steady speed.

Worked Example 3 — The parachute opens

A 75 kg skydiver falls at terminal velocity. Just after her parachute opens, the air resistance on her is 2400 N. Find the resultant force and describe her motion.

Her weight is $75 \times 9.8 = 735 N$ downwards. Before the parachute opens, at terminal velocity, air resistance is also 735 N.

Just after it opens, take up as positive. The resultant force is $2400 - 735 = 1665 N$ , upwards.

She is still moving downwards, so an upward resultant force slows her down. She does not move upwards. As she slows, air resistance falls back towards 735 N, and she reaches a new, lower terminal velocity.

Check: the direction of the resultant force tells you which way the velocity is changing, not which way the object is moving.

Worked Example 4 — A force at an angle

Aiko pulls a 20 kg suitcase along level ground at a steady speed. Her pull is 50 N, along a handle at 30° above the horizontal. Find the friction on the suitcase and the normal force from the ground.

A force is an arrow with a size and a direction, so it can be split into two parts, just like a velocity in the lecture on projectiles launched at an angle.

The 50 N pull split into a horizontal part, 50 cos 30° ≈ 43.3 N, and a vertical part, 50 sin 30° = 25 N (drawn to scale).

Horizontal part: $50 cos 30° \approx 43.3 N$ forwards. The suitcase moves at a steady speed in a straight line, so the horizontal forces balance. Friction is about 43.3 N backwards.

Vertical part: $50 sin 30° = 25 N$ upwards. The weight is $20 \times 9.8 = 196 N$ downwards. The suitcase does not rise or sink, so the vertical forces balance too: $N + 25 = 196$ , and $N = 171 N$ .

Check: the normal force is less than the weight, because Aiko's pull lifts part of the suitcase. That makes sense.

Three classic mistakes

Trap 1 — "Steady speed means no resultant force"

Only if the direction is steady too. An object moving in a circle at a steady speed has a changing velocity, so a resultant force acts on it. The first law is about constant velocity, not constant speed.

Trap 2 — "The force of the throw keeps the ball going up"

Once the ball leaves your hand, your hand no longer touches it, so it exerts no force. Ignoring air, the only force on the rising ball is its weight, downwards. The ball keeps moving up because of its velocity, not because of a force. Its weight slows it, stops it and brings it back.

Trap 3 — "At terminal velocity, there are no forces"

There are two large forces: weight and air resistance. They are equal and opposite, so the resultant force is zero. Say "zero resultant force", never "no force".

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

A lamp of weight 18 N hangs at rest from a ceiling on a single cable. What is the tension (the pull) in the cable? Explain using the first law.

Hint 1
The lamp is at rest and stays at rest. What does that tell you about the resultant force?

Hint 2
Only two forces act: the weight downwards and the cable's pull upwards.

Full answer
The lamp's velocity is not changing (it stays at zero), so by the first law the resultant force is zero. The cable's pull must balance the weight: the tension is 18 N, upwards. Check: 18 N up and 18 N down give zero.
A truck moves at a steady speed along a straight, level road. Its driving force is 4200 N. Air resistance on it is 2700 N. What is the friction from the road on its tyres, as a backward force? What would happen if the air resistance suddenly fell, for example because the wind dropped?

Hint 1
Steady speed in a straight line: the forwards and backwards forces balance.

Hint 2
Resistive forces together = 4200 N. Air resistance is part of that.

Full answer
The backward forces must add to 4200 N, so friction = $4200 - 2700 = 1500 N$ . If air resistance fell, the backward forces would be less than 4200 N. The resultant force would point forwards, so the truck would speed up until air resistance grew enough to balance again.
Mei rolls a ball across a floor. It slows down and stops. Her friend Kwame says, "That proves the first law is wrong: nothing pushed against the ball, and it stopped anyway." Is he right? Explain.

Hint 1
Was there really no force on the ball while it rolled?

Hint 2
Think about the floor and the air.

Full answer
He is not right. Friction from the floor and air resistance act backwards on the rolling ball. So there is a resultant force against the motion, and by the first law the velocity changes: the ball slows down and stops. The ball stopped because a force acted on it. On a perfectly smooth, airless surface, it would roll forever.
Diego pushes a 25 kg crate along a level floor at a steady speed. His push is 100 N, directed 20° below the horizontal (he pushes forwards and slightly down). Find the friction on the crate and the normal force from the floor. Compare the normal force with the crate's weight, and explain the difference.

Hint 1
Split the push into a horizontal part and a vertical part, as in Worked Example 4. This time the vertical part points down.

Hint 2
Horizontal: $100 cos 20°$ . Vertical: $100 sin 20°$ , downwards. Weight: $25 \times 9.8$ .

Full answer
Horizontal part: $100 cos 20° \approx 94.0 N$ forwards. The crate moves steadily, so friction ≈ 94.0 N backwards.
Vertical part: $100 sin 20° \approx 34.2 N$ downwards. Weight: $25 \times 9.8 = 245 N$ downwards. The vertical forces balance, so $N = 245 + 34.2 \approx 279.2 N$ .
The normal force is larger than the weight, because Diego's push presses the crate into the floor. In Worked Example 4 the pull lifted the suitcase, so the normal force was smaller than the weight. Check: up 279.2 N, down $245 + 34.2 = 279.2$ N.

The bigger idea

Stand in a bus that brakes suddenly, and you lurch forwards. It feels as if a force pushed you. But nothing did. The bus slowed down, and your body, by the first law, tried to keep moving at the old velocity. The "push" you feel is your own inertia, seen from inside a vehicle whose velocity is changing.

This shows that the first law holds when you measure motion from a viewpoint that is not itself speeding up, slowing down or turning. Physicists call such a viewpoint an inertial frame of reference. Inside a braking bus or a turning ride, objects seem to be pushed by forces that nothing provides. Later physics, all the way to Einstein's theories, takes this idea further. For now, measure motion relative to the ground, and the first law always holds.

Key concepts in this lesson

Newton's first law says an object stays at rest or keeps moving at constant velocity in a straight line unless a resultant force acts on it.
Motion does not need a resultant force; only a change in velocity (in speed or in direction) needs one.
When the forces on an object are balanced, the resultant force is zero, even though the individual forces may be large.
Inertia is the tendency of an object to resist changes in its motion, and mass is the measure of inertia.
A skydiver reaches terminal velocity when air resistance grows to equal her weight, so the resultant force becomes zero.